The Siegel threefold A_2 parametrizes principally polarized abelian
surfaces and has dimension three over Qbar. A point s of A_2 is
quaternionic if the full geometric rational endomorphism algebra
End^0(A_s) is an indefinite quaternion division algebra over Q. For
a fixed such algebra D the quaternionic points form a closed special
curve S_D of A_2, and the union over all D and all Hecke translates
is the quaternionic locus of A_2. A reduced irreducible Hodge-generic
algebraic curve C in A_2 over Qbar meets the quaternionic locus in
finitely many points, with a bound uniform in the quaternion algebra,
its discriminant, and the endomorphism order of A_s. This is the
quaternionic-division component of the Zilber-Pink prediction for
curves in A_2. No boundary or reduction hypothesis enters.
pdf/daw-orr-1911.05618-quantitative-reduction-theory-unlikely-intersections.pdf
pdf/pink-2005-common-generalization-andre-oort-manin-mumford-mordell-lang.pdf
An abelian surface is a complex torus of dimension two equipped
with a principal polarization, up to isomorphism.
A lattice in C^g is a Z-lattice L of full rank 2g in R^{2g} = C^g.
The quotient A = C^g / L is a compact complex Lie group, called
a complex torus. In dimension 2, A is an abelian surface when
it carries a Riemann form psi: L x L -> Z, i.e., an alternating
integer pairing on L for which the real bilinear form
(x, y) |-> psi(i x, y) is positive definite. Such a psi is a
polarization. It is principal if the matrix of psi with respect
to some (hence every) Z-basis of L has determinant 1.
The lattice description and the polarized-torus description are
equivalent, and either is used where convenient.
H^1 and its Hodge structure. Define H_Z = H^1(A, Z) = Hom(L, Z).
After tensoring with C, H_Z (x) C splits into two eigenspaces for
multiplication by i on the tangent space:
H_Z (x) C = H^{1,0} + H^{0,1}, dim H^{1,0} = dim H^{0,1} = 2.
The polarization psi gives an alternating perfect pairing on
H_Z that respects this decomposition. The Hodge structure on
H^1(A, Q) is a weight-1 Hodge structure of type (2, 2) in
dimension 4.
The reader can think of a weight-1 Hodge structure as an extra
piece of complex structure carried by a rational vector space:
a decomposition into two complex subspaces of the same dimension
that are conjugate to one another. The abelian surface A can
be recovered from this data (this is Torelli's theorem for
abelian varieties).
An endomorphism of A is a holomorphic group homomorphism u: A -> A.
Endomorphisms of A form a ring End(A). Because A = C^2 / L, any
endomorphism descends to a Z-linear map L -> L. This realizes
End(A) as a subring of End(L) = M_4(Z).
Allow rational coefficients: define
End^0(A) = End(A) (x)_Z Q,
the Q-algebra of rational endomorphisms. It is a finite
dimensional semisimple Q-algebra (dim <= 16). By a theorem of
Albert, only the following shapes occur for a surface:
Q ("no endomorphisms")
a real quadratic field (real multiplication, RM)
an indefinite quaternion division algebra over Q
(quaternionic multiplication)
M_2(Q) (E x E for non-CM E)
an indefinite quaternion algebra times M_2(Q)
a CM quartic field, or matrix over it (complex mult.)
Not needed is the full classification. What matters is:
if End^0(A) is an indefinite quaternion division algebra,
then the abelian surface A and its complex conjugate A-bar
are not isomorphic, and yet A admits "four-dimensional"
arithmetic.
The Rosati involution. The polarization psi gives a positive
definite involution "dagger" on End^0(A): for u in End^0(A),
u^dagger = psi^{-1} o u^* o psi in End^0(A),
where u^* is the adjoint of u for psi in the alternating matrix
sense. The involution is an anti-automorphism of End^0(A) over
Q, and one can check that Tr(u u^dagger) > 0 for u != 0.
Albert's positivity theorem says: every endomorphism algebra of
a principally polarized abelian variety admits such a positive
involution, unique up to a "twist" (multiplication by a positive
rational) when the variety is simple.
Definition 0.3. A QM point of A_2 is a point s such that
End^0(A_s) is an indefinite quaternion division algebra over Q.
The condition "indefinite" is forced by the positivity of the
Rosati involution in dimension 2: if D is a division quaternion
algebra and A has principally polarized QM by D, then D (x) R
must be M_2(R). Algebras with D (x) R = Hamilton quaternions
(both ramified at infinity) cannot occur as endomorphism
algebras of a principally polarized abelian variety in
characteristic 0.
The full geometric endomorphism algebra E^0(A_s) is allowed to be
endomorphism algebra of A_s (x) Qbar. The condition is on this
enlarged algebra, not on the algebra of Q-endomorphisms of any
particular model.
Note carefully what the condition excludes. It excludes:
End^0(A_s) = Q (no extra structure)
End^0(A_s) = real quadratic (RM)
End^0(A_s) = M_2(Q) (split quaternions)
End^0(A_s) = M_2(D) for D a quat. div. alg.
(products of quaternionic)
It also excludes using "partial" quaternion embedding: if
End^0(A_s) = M_2(D) for D a division quaternion algebra, the
paper does NOT count A_s as QM even though D embeds. The
reason is that in this case A_s is isogenous to a product
B x B' for some abelian surface B; the "special locus" it lies
on is much larger and gives a much easier finiteness problem,
handled by earlier work in the literature.
For every integer N >= 3 (with N chosen to avoid automorphisms),
the moduli space A_2(N) of principally polarized abelian surfaces
with full symplectic level-N structure is a fine moduli scheme,
smooth and quasiprojective, of dimension 3 over Q(zeta_N).
Passing to the coarse quotient by Sp_4(Z/N) and descending to
Qbar gives the Siegel moduli A_2.
Why dimension 3? Balance of dimensions:
Sp_4(R)-principal bundle : Siegel upper half space H_2
dim_R H_2 = 6
dim_R Sp_4(R) = 10
dim_R U(2) (stabilizer) = 4
--- 6 = 10 - 4
H_2 / Sp_4(Z) is a 3-dimensional complex orbifold.
Fine level A_2(N) is a 3-dimensional complex manifold.
Denote by
A -> A_2
the universal predominantly polarized abelian surface (over a
suitable fine level). A point t of A_2 corresponds to a fibre
A_t = A x_{A_2} {t}.
Special curves. For each indefinite quaternion division algebra
D over Q, the points where End^0(A_t) = D form a closed
algebraic curve inside A_2. These are the "quaternionic" special
curves, denoted S_D. We use the convention that S_D and its
Hecke translates (via l-isogenies) together form the part of
A_2 relevant to the main theorem.
The main theorem counts the union over all D of the S_D points,
plus the Hecke translates, uniformly across D. This uniformity
is what makes the theorem hard: the size of the quaternion's
discriminant can grow, and the level structure can vary, and
yet the count must remain finite when intersected with a fixed
Hodge-generic C.
The result can now be stated precisely.
THEOREM 0.5.1 (Finiteness of QM points on curves).
Let K be a number field. Let C be a reduced irreducible closed
algebraic curve in A_2 over Qbar that is Hodge generic, meaning
it is not contained in any proper special subvariety of A_2. Then
Sigma_QM(C) :=
{ s in C(Qbar) :
End_{Qbar}(A_s) (x)_Z Q is an indefinite quaternion
division algebra over Q }
is a finite set.
The conclusion is unconditional: neither C, nor the points s
considered, need to admit any particular reduction type, and the
quaternion algebra is allowed to vary with s.
The general conjecture. Zilber (2002) and Pink (2005)
formulated the idea of "unlikely intersections":
Let V be a Shimura variety. If an algebraic subvariety Y
of V is not contained in a proper special subvariety, then
every irreducible component of Y cap S, where S is the union
of all special subvarieties of V, is understood.
When Y is an algebraic curve C of dimension 1 inside the
3-dimensional A_2, and when C is Hodge generic, the expected
dimension of C cap S is negative. Hence C should meet S in
only finitely many points. This is exactly the kind of
assertion made by Zilber-Pink.
The prediction in our case therefore reduces to the statement:
For each Hodge-generic curve C in A_2 over Qbar, C meets
the union of all quaternionic special curves S_D in only
finitely many points, uniformly in D.
Theorem 0.5.1 is precisely this prediction, restricted to the
division-quaternionic part of the union. It does not address
the CM-by-imaginary-quadratic (almost-CM, 0-dimensional
special) part, nor the RM part, nor the split quaternionic
part. Those cases have either been addressed in the existing
literature (see below) or are strictly easier.
For a reference survey, see (Daw and Orr), (Papas), and [Pap25a,
Pap25b].
The most directly comparable theorem is:
[DawOrr22, Theorem 1.4]: if the closure of C meets the
zero-dimensional Baily-Borel boundary stratum of A_2, then
Zilber-Pink holds for C completely.
For a general C, (Daw and Orr) reduce the proof to a "large Galois
orbit" hypothesis. Their later (Daw and Orr) proves finiteness
against all special curves under the same boundary hypothesis.
(Papas) extends to higher-dimensional A_g under degeneration
hypotheses; in dimension 2, the boundary assumption is the
same as in (Daw and Orr).
(Papas) proves a height estimate in which the error is
proportional to the number of supersingular places at which the
varying point is close to a fixed centre. If that number is
bounded, finiteness follows.
Theorem 0.5.1 removes all these hypotheses. The novel
ingredients are:
(a) an interpolation theorem (Section 3) that avoids the
need for the number of supersingular proximity places
to be bounded;
(b) an auxiliary marking (Section 5) that supplies the
non-identity input to the interpolation theorem at
isogeny-exception points, without imposing any new
condition on the quaternion algebra of the target.
Two large pieces.
PIECE A. Height estimate.
For a fixed fine-level cover S of C, and fixed center t_0 in
S(Qbar) that is itself a QM point, there are constants c, k
such that every QM point t in S(Qbar) satisfies
h(t) <= c * [K(t):K]^k, (0.8.1)
where h is a fixed ample logarithmic height on the smooth
projective completion of S. The constants depend only on
the fixed data, not on t or on End^0(A_t).
PIECE B. From (0.8.1) to finiteness.
Given (0.8.1), the degree [K(t):K] of every QM point on S
is bounded. The bound on degree plus (0.8.1) gives a height
bound. By Northcott, only finitely many points s in S(Qbar)
have both degree and height bounded. This gives finiteness
on S, which descends to C because S -> C is finite.
Piece B is classical: it uses the Pila-Wilkie counting theorem
in the "path" form refined by Habegger-Pila. Section 6 does it
in full.
Piece A is where the paper's content is. An outline follows.
Local equation at a QM point. Introduce a rank-5 quadratic
local system V on S: at each t, the vector space V_t is the
space of trace-free endomorphisms of H^1(A_t) that are self-
adjoint for the polarization, modulo identity, with quadratic
form q(u) = (1/4) Tr(u^2). Section 1 constructs V and verifies
its signature is (3, 2). At a QM point t, the trace-free
Rosati-symmetric endomorphisms of A_t form a positive definite
2-plane E_t inside V_t, spanned by two actual algebraic
endomorphisms.
An analytic equation. Choose t_0, B_1, B_2 for the centered
endomorphism pair. For each QM target t with a small integral
pair P_1, P_2 (from Masser-Wustholz), let Y(a) be horizontal
transport of flat sections from t to t_0, a = x(t) a chosen
local coordinate on S with x(t_0) = 0. Then at each place at
which t is p-adically near enough to t_0, the cross-pairings
<B_i, Y(a) P_j>, 1 <= i, j <= 2,
are the traces of actual specializations of the centered and
target endomorphisms on a common good reduction, hence are
rational, of size bounded polynomially in d h.
Using the conjugate place. Taking a conjugate b = a R, where b
is a simultaneous conjugate of a and R = b / a is a further
coordinate, and applying a Frobenius lift, we transform the
initial list of rational-matrix-valued relations into a
common list of relations of the form
<B, Y(a) P> = <B', Y'(a R) P'>, R = b/a, (0.8.2)
in which the right side is a specific algebraic conjugate
of the left side. This "comparison with the conjugate"
eliminates the unknowns T_v and produces relations with fixed
coefficient size.
Proximity smallness. At most local places v of S (weighted
appropriately), the target's distance to t_0 in the v-adic
topology is measured by |a|_v. Summing -log|a|_v over
proximity places produces a "smallness sum"
S_small >= h / Q_1(d),
where Q_1 is a polynomial of controlled degree. Only a
polynomial number of groupings is needed to retain this
smallness.
Coefficient control. The common relations (0.8.2) have
coefficients that are Taylor coefficients of fixed analytic
systems, of controlled growth. Section 4 gives the precise
bounds: coefficients of order j grow like O(j) in degree and
with logarithmic height of order polynomial in d and j.
The interpolation engine. Theorem 3.2 in Section 3 takes
data of the form
- common relations E_nu(a, U) = sum_{j >= 0} a^j e_{nu,j}(U)
vanishing at the target z_* = (a_*, U_*), with the
coefficient bounds above;
- a smallness sum S_small;
- a non-identity requirement on every relevant algebraic
branch;
- controlled heights and degrees;
and produces the height bound h(t) <= c d^k. The proof is
a dimension descent: at each step, we construct a low-degree
polynomial P that vanishes at z_*, and pass to one irreducible
component of P = 0 containing z_*. The product formula
forces this vanishing once smallness dominates the height of
the target.
Non-identity. The non-identity requirement is the remaining
hard part. Suppose a branch of some W subset X has all
equations E_nu identically zero on it. The common relations
are then identities on the transport matrices Y, and the
monodromy of the pulled-back local systems enters. There are
three cases:
(i) Constant-valued relations (from archimedean places or
fixed exceptional primes). Single-argument monodromy
contradicts an identity, because the standard
representation of SO(V) on V has no invariant
vectors.
(ii) Paired relations (0.8.2) away from isogeny
correspondences. Product monodromy Sp_4 x Sp_4
contradicts an identity.
(iii) Paired relations on an isogeny correspondence. The
key case. We pass to a Frobenius quasi-isogeny on a
common reduction; the identity would force a
common sign eps = +1 or -1 on both planes. eps = +1
gives a scalar self-quasi-isogeny with impossible
multiplier. eps = -1 gives a splitting of the
integral unimodular lattice Lambda_ell that would
make a certain reduction nondegenerate; the
auxiliary marking (Section 5) ensures it is
degenerate, giving a contradiction.
(0.8.1) now follows from Theorem 3.2, and the proof of the
main theorem completes with Section 6's counting argument.
The whole proof pivots on one gadget: a rank-5 quadratic local
system V over a fine-level cover S of C, together with a 2-plane
E_t inside V_t for each QM point t. This section constructs V,
verifies its signature, identifies the 2-plane E_t, and records
the group-theoretic fact GSp_4 -> SO(V) with kernel scalars.
The remaining sections will then be pure consequences of this
structure plus arithmetic input.
Let C be the curve of Theorem 0.5.1. Pass freely to a smooth
fine-level cover, and to a dense open. Making this precise:
Lemma 1.1.1 (Finite covers with fixed comparison bounds).
Let C subset A_2 over Qbar be a reduced irreducible curve. There
is a finite surjective morphism pi: S -> C from a smooth
geometrically connected curve S over a number field, together
with a principally polarized abelian scheme f: A -> S whose
coarse map to A_2 agrees with pi. The map pi may be constructed
by normalization followed by a full symplectic level N with
N >= 3.
If C contains infinitely many QM points, then S does. Deleting
finitely many points of S, or making further finite surjective
covers and selecting a dominating connected component, preserves
infinitude. Once all fixed data are defined over K, degrees and
ample logarithmic heights on the smooth projective completions
satisfy:
[K(pi(t)):K] <= [K(t):K] <= D [K(pi(t)):K] (1.1.a)
c^{-1} h_C(pi(t)) - c <= h_S(t) <= c h_C(pi(t)) + c (1.1.b)
for fixed positive constants D, c independent of t.
Proof. The existence of the fine-level moduli scheme A_2(N)
together with its universal abelian scheme is standard
[Chai85, Chapter I, Theorem 1.4]. Its forgetful map to the
moduli space A_2 is finite (change of level). Normalize a
component of the pullback over C; normal curves in
characteristic zero are smooth, so this gives S smooth. Everything
descends to a number field.
A finite map has finite fibres, so an infinite subset of C lifts
to infinitely many points of S. The degree comparison (1.1.a)
comes from the fact that the degree over C is bounded by the
degree of the finite map (with a finite correction at the
finitely many singular points). The height comparison (1.1.b)
follows from functoriality of heights: amplitudes pull back to
amplitudes compatibly, and any two ample divisors on a fixed
projective curve bound each other's heights linearly, with
bounded error. QED
From this point on, fix one such (S, A -> S) with the level N
large enough that no point of S has nontrivial symplectic
automorphism. Work with S, not C, and add the finite deleted
points back at the very end. Every "QM point" of S below means a
lift of a QM point of C. By Lemma 1.1.1 there are infinitely
many such lifts if and only if there are infinitely many QM
points on C.
Assume from now on, for the sake of contradiction:
S has infinitely many QM points. (Hyp-inf)
Fix notation. Let
H_Z = R^1 f_* Z on S(C), H = H_Z (x)_Z Q.
The alternating polarization psi of the abelian scheme gives an
alternating perfect pairing
psi: H x H -> Q,
compatible with the variation of Hodge structure on H (weight 1,
Hodge numbers (1, 1) each in dimensions 2, 2, total rank 4).
For an endomorphism T: H -> H, define its adjoint T^dagger by
psi(T^dagger x, y) = psi(x, T y), all x, y in H.
T^dagger is again a rational endomorphism of H. The map T -> T^dagger
is a Q-linear involution of End(H). Its fixed subspace is the
space of "self-adjoint" (or "Hermitian") endomorphisms for psi.
Definition 1.2.1 (operator system). Put
V = { T in End(H) : T^dagger = T, Tr(T) = 0 }, (1.2.a)
<T, U> = (1/4) Tr(TU), q(T) = <T, T>. (1.2.b)
V is the local system underlying the construction. Here and
below Tr denotes the cohomological trace on H^1, so that Tr(1) = 4.
Lemma 1.2.2 (V is a rank-5 quadratic local system).
(a) V has rank 5 over Q.
(b) <., .> is symmetric and nondegenerate on V.
(c) q has signature (3, 2) as a real quadratic form on V (x) R.
(d) V carries a weight-zero Hodge structure with Hodge numbers
h^{1,-1} = 1, h^{0,0} = 3, h^{-1,1} = 1.
(e) V is isomorphic to the primitive part of Lambda^2 H (1)
as Q-local systems with Hodge structure.
(f) The natural integral lattice
V_Z = V intersection End(H_Z)
is integral and unimodular after inverting 2.
Proof. Part (a) and part (b) are the content of the calculation
below. Part (d) and part (e) follow from the same calculation
combined with the standard description of Lambda^2 H (1) (see
below). Part (f) is checked in the given basis.
Work on a single fibre, in a symplectic basis e_1, e_2, f_1, f_2
of H such that
psi(e_i, f_j) = delta_{ij}, psi(e_i, e_j) = psi(f_i, f_j) = 0.
Matrix of psi is then the block form
J_4 = [ 0 I_2 ] ,
[ -I_2 0 ]
with I_2 the 2x2 identity. Write the matrix of T as a 2x2 block
matrix [[A, B], [C, D]] with A, B, C, D in M_2(Q). The
self-adjointness condition T^t J_4 = J_4 T expands to
A^t = D, C^t = -C, B^t = -B.
Trace is Tr(T) = tr(A) + tr(D) = 2 tr(A), where tr is the 2x2
matrix trace. Trace-free therefore means tr(A) = 0.
Write
A = [ a b ] , B = beta J_2,
[ c -a ] C = gamma J_2,
J_2 = [ 0 1 ]
[ -1 0 ],
with scalars a, b, c, beta, gamma in Q. Then
T = [ [a b; c -a] beta J_2 ]
[ gamma J_2 [a c; b -a] ]
(the 2x2 blocks filled in). This T depends on exactly 5 rational
parameters. Hence dim_Q V = 5. Computing T^2 as a 4x4 matrix
gives
T^2 = (a^2 + bc - beta gamma) I_4.
Hence q(T) = (1/4) Tr(T^2) = a^2 + bc - beta gamma.
Change coordinates: put
y = b + c, z = b - c, so bc = (y^2 - z^2)/4,
u = beta + gamma, v = beta - gamma,
so -beta gamma = (v^2 - u^2)/4.
Then
q = a^2 + (y^2)/4 - (z^2)/4
+ (v^2)/4 - (u^2)/4.
The form has three positive square terms (a^2, y^2, v^2) and two
negative ones (z^2, u^2), each coefficient 1/4 rescaling one
coordinate. Hence signature (3, 2), proving (c).
For (d): write H (x) C = H^{1,0} + H^{0,1}, each of dimension 2.
Then Lambda^2 H (x) C decomposes as
Lambda^2 H (x) C = (Lambda^2 H^{1,0})
+ (H^{1,0} (x) H^{0,1})
+ (Lambda^2 H^{0,1}),
with respective dimensions 1, 4, 1. Tensor by the Tate twist Q(1)
(which shifts types by +1) and take self-adjoint, trace-free
part. The three pieces of Lambda^2 H (x) C sit in types (2, 0),
(1, 1), (0, 2). After (1) they sit in (3, 1), (2, 2), (1, 3),
weights 4 = 2 + 2. The middle piece, being the self-adjoint part,
is where we look. Take the primitive part with respect to psi:
that is, require the 2-form to pair with psi to a nondegenerate
form; equivalently, remove the psi-direction. In the middle piece
dim 4, one direction (psi itself) is the "trace". The remaining
3 pieces are the primitive (0, 0)-type part; the two remaining
Hodge types come from the outer pieces plus the middle
"anti-trace" pieces. Concretely:
- Outer piece Lambda^2 H^{1,0} (twisted) contributes type
(1, -1) after removing the "trace" component, dimension 1.
- Middle piece H^{1,0} (x) H^{0,1} (twisted) has type
(0, 0), dimension 4 minus the trace = 3.
- Outer piece Lambda^2 H^{0,1} (twisted) contributes type
(-1, 1), dimension 1.
Sum 1 + 3 + 1 = 5. Hence V has Hodge numbers (1, 3, 1) in
types (1, -1), (0, 0), (-1, 1), proving (d).
Part (e) is the composed statement: the self-adjoint trace-free
operators are exactly the primitive Lambda^2 H (1) bundle, because
the alternating 2-form psi (T x, y) records T, and the trace-zero
condition records primitivity. This is completely standard
[Mum70, Section 24]. Part (f): the intersection of End(H_Z) with
the self-adjoint trace-free condition is a lattice in V. Its Gram
matrix with respect to the basis {diag-block, E_12, E_21, beta J_2,
gamma J_2} is integral and unimodular after inverting 2, so
det(V_Z) is a power of 2. QED
Let t in S(Qbar) be a QM point, with abelian surface A_t and
polarization lambda_t. Write D = End^0(A_t) for its endomorphism
algebra (D is by assumption an indefinite quaternion division
algebra over Q).
The Rosati involution "dagger" associated to lambda_t is a
positive-definite involution of D. Restricted to the 4-dim
Q-algebra D, its fixed subspace has dimension 3 (this is Albert's
theory: for a positive involution on a quaternion division
algebra, the fixed part is the "twisted skew" part, isomorphic to
a trace-0 subspace of M_2(R), which is 3-dimensional).
The cohomological trace Tr = 4 * (matrix trace) is a surjective
homomorphism D -> Q. Its kernel has dimension 3 (since D is
4-dim over Q). Inside that kernel, the self-adjoint subspace is
2-dimensional. This is the relevant two-dimensional space.
Lemma 1.3.1 (QM 2-plane).
Let t be a QM point with algebra D = End^0(A_t). Define
E_t = { u in D : u^dagger = u, Tr(u) = 0 } in V_t. (1.3.a)
Then:
(a) E_t is a 2-dimensional Q-subspace of V_t.
(b) q restricted to E_t (x) R is positive definite.
(c) E_t is spanned by two actual integral endomorphisms of A_t
(after a fixed finite extension of K(t)).
(d) The intersection L_t = E_t intersection V_Z, t is saturated
in V_Z, t.
(e) A chosen spanning pair of integral endomorphisms of E_t
need not generate L_t; it may span a sublattice of finite
index.
Proof. (a) is the dimension count above.
(b) For u in E_t (x) R, viewed inside D (x) R = M_2(R) with
dagger = transpose (in a suitable basis), there is
q(u) = (1/4) Tr(u u^dagger) = (1/4) Tr(u^2).
For u symmetric trace-zero in M_2(R), u^2 is positive-semidefinite
with trace = 2 tr(u^2) > 0 for u != 0, because u has real
eigenvalues summing to 0 in each 2x2 block. Concretely if u has
eigenvalues mu, -mu, then Tr(u^2) = 2 mu^2 > 0. Hence q is
positive definite on E_t (x) R.
(c) Rational elements of D can be cleared to integrality: given
Q-basis vectors v_1, v_2 of E_t, multiply each by a positive
integer to get integral elements (this preserves trace-zero and
dagger-self-adjointness). The scale only enlarges the field of
definition by a bounded factor.
(d) An intersection of a Z-lattice with a Q-subspace is saturated
by definition (if n v in Lambda and v in E_t (x) Q, then v in
Lambda because Lambda is a lattice and E_t (x) Q is a
Q-subspace). So L_t = E_t intersection Lambda_t is saturated.
(e) The pair (v_1, v_2) of (c) is Z-independent and Q-spans E_t,
but its Z-span is only a sub-lattice of L_t, of some finite index
(possibly > 1). This matters in Section 5. Adding new products
v_i v_j necessarily lands back in D but not necessarily in
L_t (x) Q.
QED
The pair (v_1, v_2) will be called the "target endomorphism pair".
In symbols write (P_1, P_2) for the de Rham coordinates of
(v_1, v_2), once the de Rham trivialization of V has been set up
in Section 1.5. The pair (B_1, B_2) will be the analogous pair at
the fixed centre t_0 (Section 4).
Let GSp_4 = GSp(H, psi) = GSp_4(H) be the general symplectic
group of (H, psi), i.e., the group of g in GL(H) with
psi(g x, g y) = c psi(x, y) for all x, y,
where c = c(g) is a nonzero rational (the multiplier of g).
Its subgroup Sp_4 = Sp(H, psi) is the c = 1 case.
Lemma 1.4.1 (GSp_4 -> SO_5).
The conjugation map
rho(g)(T) = g T g^{-1} : V -> V, g in GSp_4, (1.4.a)
is a morphism of Q-algebraic groups, and:
(i) Image of rho is exactly SO(V, q).
(ii) Kernel of rho is the scalar torus G_m, acting trivially
on V; on the adjoint group PSp_4 = GSp_4 / G_m, rho
is an isomorphism.
(iii) Restricted to Sp_4, rho is the spin covering
Sp_4 -> SO_5, with kernel {+I_4, -I_4}.
(iv) Everything is defined over Q.
Proof.
Well-definedness. Let g in GSp_4 with multiplier c. Then
g^dagger = c g^{-1} (adjointness: psi(g^dagger x, y) = psi(x, gy)
and psi(gx, g y) = c psi(x, y) together imply g^dagger g = c I, so
g^dagger = c g^{-1}). For T in V:
(g T g^{-1})^dagger
= (g^{-1})^dagger T^dagger g^dagger
= (c^{-1} g) T (c g^{-1})
= g T g^{-1},
using T^dagger = T. So g T g^{-1} is self-adjoint. Its trace is
Tr(g T g^{-1}) = Tr(T) = 0. So g T g^{-1} is again in V.
Preservation of q: q(g T g^{-1}) = (1/4) Tr((g T g^{-1})^2)
= (1/4) Tr(g T^2 g^{-1}) = (1/4) Tr(T^2) = q(T). So rho(g) is an
isometry of (V, q), and rho is a group homomorphism GSp_4 -> O(V, q).
Determinant. Since GSp_4 is connected (as an algebraic group over
Q) and the determinant is a continuous character, rho(GSp_4) sits
inside the identity component SO(V, q) of O(V, q). So
rho(GSp_4) subset SO(V, q).
Surjectivity onto SO_5. On Lie algebras rho induces
sp_4 -> so(V, q), X |-> [X, .], (1.4.b)
both sides of dimension 10. The image is nonzero (take X with an
off-diagonal entry that does not commute with T = beta J_2). Since
so_5 is simple, the image is all of so_5. So rho(GSp_4) contains
SO_5.
Actually the last step is a dimension count: the differential of
rho is surjective at the identity, so rho(GSp_4) has dimension at
least dim SO_5 = 10 inside the 10-dimensional target. Hence
rho(GSp_4) is the full SO_5.
Kernel. The scalars cI_4 act trivially on V: (cI) T (cI)^{-1} = T.
So G_m subset ker rho. From the surjectivity
rho: GSp_4 -> SO_5, ker rho has dimension dim GSp_4 - dim SO_5
= 11 - 10 = 1. G_m has dimension 1 and is contained in ker rho.
Hence ker rho = G_m.
Spin covering. Restricted to Sp_4, dim Sp_4 = 10 = dim SO_5 and
image is all of SO_5. Kernel is Sp_4 cap G_m = {+I_4, -I_4}.
So Sp_4 -> SO_5 is a 2-fold covering, and Sp_4 is simply connected
(as an algebraic group); this identifies Sp_4 = Spin_5.
Over Q. V, q, and the map rho are all defined over Q (the
constructions use only Q-rational data: the symplectic form and the
trace). QED
Remark 1.4.2. The "spin" identification is important in Section 5
where it is necessary to lift elements of SO(V, q)(k) (over a finite field k)
to Sp_4(k). Lemma 1.4.1(iii) says every mod-l change of
symplectic marking is available in the image of the spin covering.
The operator system V has a Betti realization (on H_Z), an algebraic
de Rham realization V_{dR}, and, for each prime l, an etale
realization V_l. All three are rank-5 local systems with the same
quadratic form q. We use them interchangeably, indicating the
realization only when needed.
De Rham realization. Let H_{dR} = R^1 f_* Omega^*_{A/S}, with its
Gauss-Manin connection. The algebraic-analytic de Rham comparison
(given by [Grothendieck1966, Katz1970]) identifies its horizontal
sections with H_Z (x) C. The trace-free self-adjoint operators
inside End(H_{dR}) form V_{dR}, with the same quadratic form q as
V.
Endomorphisms. An algebraic endomorphism alpha of A_t has
Betti, de Rham, and etale realizations alpha_B, alpha_{dR}, alpha_l
on H, H_{dR}, H_l. These are compatible: they give the same
operator in V (in the appropriate realization), and the pairings
Tr(alpha beta) are the same rational numbers in every realization.
Isogenies. Let phi: A -> A' be a polarized isogeny with
phi^* lambda' = m lambda for a positive integer m (here lambda,
lambda' are the principal polarizations of A and A'). Because
both polarizations are principal, deg phi = m^2. We call m the
multiplier of phi.
Cohomological conventions. On Betti (resp. de Rham) cohomology,
phi induces phi^*: H^1(A') -> H^1(A) with multiplier m and reverse
direction. To get covariance, set:
T_phi = m (phi^*)^{-1} : H^1(A) -> H^1(A'), (1.5.a)
so T_phi sends A's cohomology to A's. By construction the
multiplier of T_phi is again m. The operator transport
corresponding to phi is:
I_phi : End(H^1(A)) -> End(H^1(A')), (1.5.b)
I_phi(u) = T_phi u T_phi^{-1}.
It satisfies:
I_phi(alpha) = phi alpha phi^{-1} for alpha in End^0(A). (1.5.c)
Composition of isogenies gives composition of I's: I_{phi o psi}
= I_phi o I_psi.
Quasi-isogenies and Frobenius. Conventions extend to polarized
quasi-isogenies (rational multiplier m in Q_{>0}). In particular,
an inverse isogeny of multiplier m composed with a residue
Frobenius of multiplier p (on some good reduction) has, on the
covariant weight-one realization above, multiplier p / m. This
is the convention used in Section 5 for the "Frobenius
quasi-isogeny" u = phi_0^{-1} F.
Discussion. The reader may find the convention T_phi = m (phi^*)^{-1}
suspicious. A check: apply T_phi to the polarization form psi_A
of A. Because phi^* lambda' = m lambda, and both are principal
polarizations, the form T_phi psi_A T_phi^dagger is m psi_{A'}, so
T_phi relates the two symplectic structures up to a scalar, as
required of a morphism GSp(A) -> GSp(A'). In particular I_phi is
a morphism of the associated Shimura data. So the convention is
correct up to the standard "the isogeny is determined by its
behaviour on a fixed polarization-class".
Fix a point t of S(C). The Hodge structure of V at t (as in Lemma
1.2.2(d)) has pieces of types (1, -1), (0, 0), (-1, 1). If we
trivialize V over a simply connected open of S(C), these pieces
become subspaces V^{1,-1}(t), V^{0,0}(t), V^{-1,1}(t) of the fixed
complex vector space V_C = V (x) C:
V_C = V^{1,-1}(t) + V^{0,0}(t) + V^{-1,1}(t), (1.6.a)
with dimensions 1, 3, 1.
Definition 1.6.1 (Period line). The period line at t is the 1-
dimensional (complex) subspace Phi(t) = V^{1,-1}(t) of V_C. Its
projective class [Phi(t)] in P(V_C) = P^4(C) is the period point
of the operator system V.
Proposition 1.6.2 (Period quadric).
Define the open quadric
Q_V = { [z] in P(V_C) : q(z) = 0 }. (1.6.b)
Then:
(a) Q_V is a smooth projective quadric of dimension 3 in P^4.
(b) For every t, the period point [Phi(t)] lies on Q_V.
(c) If z in V_C is a real vector of type (0, 0) at t, then
<z, Phi(t)> = 0. Conversely, every real vector w in V_C with
<w, Phi(t)> = 0 is of type (0, 0) after Hodge decomposition.
(d) If t is a QM point and E_t is the QM plane of Lemma 1.3.1,
then E_t is contained in the (0, 0)-type part of V_t. In
particular <u, Phi(t)> = 0 for every u in E_t.
Proof. (a) The quadric q = 0 in P^4 is a smooth smooth projective
variety of dimension 3, because q is a nondegenerate form of
signature (3, 2). (The real points form a compact 3-fold.)
(b) Let z in V_C be a representative of Phi(t) = V^{1,-1}(t).
By properties of Hodge decomposition, q(z) = <z, z> = 0, because
the trace form pairs only pieces of complementary type and (1, -1)
+ (1, -1) = (2, -2) != (0, 0). So [z] lies on the quadric.
(c) If z is a real vector of type (0, 0), then z is in V^{0,0}(t)
in V_C (the real (0, 0) part). The trace form pairs V^{0,0} with
V^{1,-1} to zero because (0, 0) + (1, -1) = (1, -1) which is not
(0, 0) (no such type exists in a weight-0 structure except (0, 0)
itself, and V^{1,-1} is not (0, 0)). So <z, Phi(t)> = 0.
Conversely, if w in V_C is real and <w, Phi(t)> = 0, then by
decomposing w into pieces of type (1, -1), (0, 0), (-1, 1), the
pairing with a nonzero vector of type (1, -1) requires that the
(1, -1) and (-1, 1) pieces of w vanish, leaving w of type (0, 0).
(d) Every endomorphism of A_t is an H^1-map of type (0, 0) with
respect to the Hodge structure of H. Hence its realization in V_t
lies in the (0, 0) part. So E_t subset V^{0,0}(t). Applying (c),
every u in E_t is orthogonal to Phi(t). QED
Remarks on Q_V.
* Q_V has real signature (3, 2) and complex dimension 3. Its
real points form a 3-dimensional real quadric = SO(3,2)/K.
Monodromy (Section 2) acts on Q_V by the connected component
SO(V, q) of the orthogonal group.
* The passage from V to Q_V is the classical "period map":
the pair (V, q) and the Hodge filtration of V together give
a map S(C) -> Q_V. The period map is holomorphic and, for
Hodge generic S, Zariski dense. Section 6 uses this.
* For a QM point t, E_t is a positive 2-plane on the quadric;
its orthogonal complement in V_C is a 3-dimensional (2, -1)
form-space whose isotropic lines form a conic: the conic
consisting of points of Q_V that are orthogonal to E_t.
This conic is a 1-dimensional subvariety of Q_V, and the
period Phi(t) lies on it by (d). Section 6 exploits this
family of conics to bound the period image if the
coordinates of E_t are small.
* Not needed is the finer analysis of the "primitive part"
as a GSp_4-module. The dimension count and the group map
GSp_4 -> SO_5 suffice.
The local information from Section 1 gives, at every QM point t,
a positive-definite 2-plane E_t inside V_t. To prove non-identity of certain local equations, it is necessary to know that monodromy "twists" those planes around
without invariant vectors. This section establishes the relevant
monodromy statements.
Also proven is (Proposition 2.3.1) that between two fixed Hodge-
generic curves, only finitely many dominating isogeny
correspondences exist. This finiteness is critical: without it,
Section 5 would need to handle infinitely many "exceptional" cases
of the auxiliary marking argument.
Let S = S_1 be a smooth fine-level curve mapped to a Hodge-generic
C subset A_2. Let H = H^1(A, Q) be the weight-1 local system on
S(C), with its quadratic variation V (Section 1).
Proposition 2.1.1 (Single monodromy).
The connected monodromy group of H on a simply connected open of
S(C) equals Sp_4 = Sp(H, psi). The connected monodromy group of
V equals SO(V, q) = SO_5.
The same remains true after pullback by any dominant morphism
W -> S from a resolution W of an irreducible complex algebraic
variety.
Proof. The generic Mumford-Tate group MT of the Hodge structure H
(equivalently, of the Hodge structure V) is the group GSp_4 =
GSp(H, psi). This is by definition of Hodge-generic: a curve in
A_2 that is not contained in any proper special subvariety has
full generic Mumford-Tate group (compare (Deligne) for the
oriented statement, or [Andre1992, Section 6] for its use for
Siegel varieties).
Andre's normality theorem [Andre1992, Section 5, Theorem 1] for a
polarizable variation of Hodge structure over a smooth complex
algebraic base: the connected monodromy group is a normal
subgroup of the derived group of the generic Mumford-Tate group.
The derived group of GSp_4 is Sp_4.
So the connected monodromy H_conn on H is a normal subgroup of
Sp_4. Since Sp_4 is Q-simple, its only normal subgroups are
{1}, its center {+I, -I}, and itself. The center {+I, -I} acts
trivially on H (it is scalar), so on Sp_4 acts trivially on
H, meaning H_conn cannot equal {+I, -I} unless it acts as identity.
Since H_conn is a subgroup of GL(H) and {+I, -I} maps to {+I},
H_conn must be a subgroup of PSp_4 acting through Sp_4.
More carefully: H_conn is a connected algebraic group, hence does
not have +-I as a connected subgroup. So the possibilities are
H_conn = 1 or H_conn = Sp_4.
If H_conn = 1, the monodromy is finite. Then by the theorem of
the fixed part [Andre1992, Section 6, Remark 1], after passing to
a finite cover the local system becomes constant. But the moduli
map S -> A_2 would then be constant (varying the abelian surface
A_t changes the polarized Hodge structure H_t continuously, so a
locally constant (H_t) forces t to be locally constant on the
finite cover, hence on S). Since S is a curve covering a curve,
this is a contradiction.
Consequently H_conn = Sp_4 on H. Pushing through rho:
GSp_4 -> SO_5 gives V-monodromy = SO_5.
The same reasoning applies to any dominant pullback: the moduli
map remains nonconstant, and the generic Mumford-Tate group is
preserved under pullback along a dominant map.
QED
Remark 2.1.2. The result "V-monodromy = SO_5" is stronger than
needed in Section 5: all that is used is that the standard
representation of SO_5 has no invariant vector. That is
immediate from irreducibility of the standard representation of
SO_5 (which is a general fact about irreducible representations of
simple groups).
Now take two smooth Hodge-generic fine-level curves S_1, S_2
contained in A_2. Let W be an irreducible complex algebraic
variety, and suppose there are two dominant morphisms
pi_1: W -> S_1, pi_2: W -> S_2.
Write H^{(i)} = pi_i^* H^{(i)}, i = 1, 2, for the two pulled-back
weight-1 local systems on W. Their sum H^{(1)} + H^{(2)} is a
weight-1 polarizable VHS of rank 8.
Proposition 2.2.1 (Product monodromy).
One of the following holds:
(i) The connected monodromy group of H^{(1)} + H^{(2)} on W
is Sp_4 x Sp_4 (product of the monodromy of the two
factors).
(ii) The image of W in S_1 x S_2 is contained in a closed
algebraic curve, and that curve is a component of a
polarized isogeny correspondence: there is a fixed
element g in GSp_4(Q) with positive multiplier mu such
that the polarized Hodge structures of the two fibres are
related by (multiplication by mu and) g.
In case (ii) g is a rational positive projective symplectic
similitude carrying (a lift of) S_1 onto (a lift of) S_2.
Proof. Let H be the connected monodromy of H^{(1)} + H^{(2)},
projecting onto Sp_4 in each factor by Proposition 2.1.1. Let
G = PGSp_4 = PSp_4 be the adjoint form.
Reduction to adjoint form. In the adjoint group G = PGSp_4, let
H' subset G x G be the image of H under the adjoint map
GSp_4 x GSp_4 -> G x G. Both projections of H' onto the two
G-factors are surjective, because the original monodromy H
projects onto Sp_4 in both factors and Sp_4 -> G is surjective.
Goursat's lemma. By Goursat's lemma for algebraic groups (as
applied to groups with surjective projections, (Goursat) or
[Lang, IV]), H' is either the full product G x G, or the graph
of an automorphism phi: G -> G.
Product case. If H' = G x G, then H has image the full adjoint
product. Given that H itself is a subgroup of GSp_4 x GSp_4
projecting onto Sp_4 in each factor with center contained in the
central G_m x G_m of the product, it follows H = Sp_4 x Sp_4.
This is case (i).
Graph case. Suppose H' is the graph of an automorphism
phi: G -> G.
Since G is the split adjoint group of type C_2 (equivalently B_2,
since PGSp_4 = SO_5), its Dynkin diagram is
o == o (C_2)
without automorphisms. Hence every automorphism of G = SO_5 is
inner. So phi is conjugation by some element g_0 in G(Q) (the
Q-rationality is because the automorphism is defined over Q: the
Diophantine structure of the two monodromy systems is Q-rational).
Lifting to GSp_4. The exact sequence
1 -> G_m -> GSp_4 -> G -> 1 (2.2.a)
of Q-algebraic groups has a Q-point g in GSp_4(Q) mapping to g_0,
by Hilbert 90 (H^1(Q, G_m) = 0 since G_m is a divisible and
connected group). Then the graph of phi is the conjugate by g of
the diagonal (g_0-conjugation of the diagonal is still diagonal
when we conjugate the second factor by g: the resulting graph
becomes the diagonal in the twisted coordinates).
Normality in M. Let M be the generic Mumford-Tate group of
H^{(1)} + H^{(2)} on the product. Then M is a subgroup of
GSp_4 x GSp_4 whose adjoint image is contained in the normalizer
of the diagonal in G x G (since the diagonal H' = graph(phi) is
normal in Mi' by Andre's normality). The normalizer of the
diagonal in G x G is the diagonal itself: if (g_1, g_2) maps
the diagonal to itself under (h, h) -> (g_1 h g_1^{-1}, g_2 h
g_2^{-1}), then g_1 h g_1^{-1} = g_2 h g_2^{-1}, i.e.,
g_2^{-1} g_1 h = h g_2^{-1} g_1 for all h in G, so g_2^{-1}g_1
is central in G; hence (g_1, g_2) equals (g_2 h_0, g_2) where
h_0 is central, and modulo center g_1 = g_2.
Conclusion. Hence the adjoint image of M is the diagonal. The
two Hodge homomorphisms
Res_{C/R} G_m -> G x G,
one for H^{(1)}, one for H^{(2)}, agree after passage to the
adjoint. Their discrepancy on the weight-one spaces is a real
algebraic character
Res_{C/R} G_m -> the scalar G_m in GSp_4.
Every such character is a power of the norm N: C^* -> R^*.
The two weights are equal (both are 1), so restricting to R^*
(scalars from R) this power must be zero; hence the discrepancy is
trivial. This means the two Hodge structures on H^{(1)},
H^{(2)} agree after multiplication by g.
Fixed period relation. The above Hodge agreement holds at very
general points of W. Holomorphicity of the period map extends
the resulting period relation to all of W by analytic continuation.
Isogeny. The relation between H^{(1)} and H^{(2)} (after
multiplication by mu and g) is a rational similitude
g: H^1(A_1) -> H^1(A_2) with positive multiplier mu. The
multiplier is positive because it compares the positive definite
forms induced by the polarizations and complex structures; both
sides are positive definite, so the scalar ratio is positive.
From the linear map g on H^1, we pass to an abelian surface
isogeny: the dual map on homology is F = mu (g^{-1})^vee: H_1(A_1)
-> H_1(A_2), which commutes with complex structures, and after
multiplying by some positive integer n, becomes integral. Then
phi = nF: A_1 -> A_2 is a holomorphic isogeny of multiplier
m = n^2 mu (positive integer), extending the correspondence
structure.
The image of W in the product S_1 x S_2 therefore lies in the
closure of the Hecke correspondence attached to g. Since Hecke
correspondences are closed, the image lies in a closed algebraic
curve inside S_1 x S_2. QED
Remark 2.2.2. Case (ii) is "degenerate": the two curves are
isogenous. This is the critical case for Section 5, where the
main difficulty will be handling the Frobenius quasi-isogeny that
arises on such a correspondence.
Remark 2.2.3. In case (i), there is H_conn = Sp_4 x Sp_4. In
Section 5 this will be used to rule out a certain functional
identity: product monodromy means the standard representation of
Sp_4 x Sp_4 has no invariant vector, hence neither does the
combined operator system, preventing any common trace relation
from being constant under monodromy.
Fix, once and for all in this subsection, a finite list of
Hodge-generic curves C_1, ..., C_n in A_2 over Qbar. For each
i we select a smooth fine-level curve S_i -> C_i as in Lemma
1.1.1. Our goal is to show that only finitely many "dominating
isogeny correspondences" exist between these curves.
Definition 2.3.1. A dominating isogeny correspondence between
S_i and S_j is a closed irreducible algebraic curve Y subset
S_i x S_j whose two projections to S_i and S_j are both
dominant, such that for every point (s, t) in Y at the generic
level there is a polarized isogeny phi: A_s -> A_t whose
underlying map is fixed rational-symplectic (as in Section
1.5).
Concretely, Y is (a component of) the closure of the set of
pairs (s, t) with H^1(A_s) isomorphic to g H^1(A_t) for some
fixed g in GSp_4(Q).
The aim of this subsection is:
Proposition 2.3.2 (Finite correspondences).
Fix C_1, ..., C_n and their fine covers S_1, ..., S_n.
Then:
(a) For every ordered pair (i, j), there are finitely many
rational positive projective symplectic similitudes
g in GSp_4(Q) / Q^* with positive multiplier that carry
(a lift of) S_i onto (a lift of) S_j.
(b) Every dominating polarized isogeny correspondence
between S_i and S_j arises this way, so the set of all
such dominates a finite union of closed subvarieties of
S_i x S_j.
(c) There is a fixed finite set M = M(C_1, ..., C_n) of
positive integers such that every dominating isogeny
correspondence between some S_i and some S_j carries
an isogeny of one of the multipliers in M.
Every such isogeny has degree m^2.
(d) Refining the level, replacing S_i by a further fixed
finite cover, does not enlarge the required set M.
Proof. Proceed in four steps.
Step 1 (Real stabilizer is discrete). Let H_2 be Siegel space
and G_R = PGSp_4(R)^+ its connected group of holomorphic
isometries (which is PSp_4(R), connected). Fix a torsion-free
principal congruence subgroup Γ of Sp_4(Z), at a fine level.
For a curve S_i in our list, let X be an irreducible analytic
component of the full inverse image of S_i in H_2. It is a
closed analytic curve (of dimension 1 over C).
Let Γ_X = {γ in Γ : γ X = X}. This contains the deck group
of the lift over the smooth part of S_i, which is the image of
the monodromy of the normalization of S_i.
By Proposition 2.1.1, the monodromy is Zariski dense in G_R.
So Γ_X is Zariski dense in G_R.
Let H_X = {g in G_R : g X = X}. Claim: H_X is discrete.
- H_X is closed. Suppose g_k in H_X with g_k -> g. By
closedness of X, g X ⊆ X. Similarly for g_k^{-1} -> g^{-1}:
g^{-1} X ⊆ X, hence X ⊆ g X. So g X = X.
- H_X is a Lie subgroup of G_R (being closed).
- Its Lie algebra L = Lie H_X is invariant under Γ_X. But
Γ_X is Zariski dense in G_R, so L is invariant under G_R,
i.e., L is an ideal of sp_4(R) = Lie G_R.
- sp_4(R) is simple (as a real Lie algebra). So L = 0 or
L = sp_4(R).
- If L = sp_4(R), then H_X contains the full identity
component of G_R, which by connectedness of G_R is all of
G_R. So H_X = G_R.
- But G_R acts transitively on H_2, so H_X acts transitively
on H_2 = G_R / K_max. So X = H_X . z for any z in X
equals H_2, contradicting dim X = 1 (over C) vs dim H_2 = 3.
- Hence L = 0, H_X is discrete.
Step 2 (Finite area). Next we check that Γ_X \ X has finite
area for the restriction of the invariant Riemannian metric on H_2.
- Let S_X = normalization of Γ_X \ X. Since Γ_X contains the
deck transformations of X -> (S_i in the fine level), S_X is
a finite cover of S_i.
- Form Y = S_X x_{Γ\H_2} H_2. This is a finite morphism to
H_2 (base change of the finite map S_X -> Γ\H_2). Its image
is the full inverse image of S_X in H_2.
- Y -> S_X is a covering map (pullback of a covering), so Y is
smooth. Above the smooth locus of S_X, the map Y -> image
is an isomorphism (since the image is the full inverse image
and S_X is already normalized).
- So on the smooth part of S_X, the pullback metric from H_2
equals the metric on X = image.
- At punctures of S_X, bound the pullback metric. Any
holomorphic disk f: D -> H_2 contracts the invariant metric
up to a fixed constant, relative to the Poincare metric on
D. To see this, realize H_2 as a bounded domain, apply
disk automorphisms to move the image of zero to a fixed
point, and use the Cauchy derivative estimate in the bounded
coordinates.
- Apply on universal cover of a punctured disk 0 < |u| < ε.
The descended upper bound on Γ_X \ X is a constant multiple
of
|du|^2 / (|u|^2 log^2(1/|u|)).
This has finite integral on the punctured disk.
- On the compact part of X minus punctures, the period map is
holomorphic with finite area.
- Hence Γ_X \ X has finite total area. QED (Step 2).
Step 3 (Finite index). Claim:
[H_X : Γ_X] < ∞. (2.3.a)
- G_R acts properly on H_2 = G_R / K_max (K_max compact). So
the discrete group H_X acts properly on X, with finite point
stabilizers (in X and in H_X).
- Choose a regular point z in X and a small disk D contained
in X, positive area, H_{X,z}-invariant, such that g D ∩ D =
empty for every g not in H_{X,z}.
- For each double coset in Γ_X \ H_X / H_{X,z}, the union of
its translates of D in Γ_X \ X is disjoint from those of any
other double coset.
- Each translate embeds as a sub-disk of Γ_X \ X: an
identification of two translates would give a nonidentity
element of Γ_X conjugate to an element of the finite group
H_{X,z}. But Γ_X is torsion-free (we chose the congruence
subgroup torsion-free), contradiction.
- All translates have the same positive area. Finite total
area (Step 2) bounds the number of double cosets.
- So #(Γ_X \ H_X / H_{X,z}) finite, and #H_{X,z} finite,
giving (2.3.a).
Step 4 (Finitely many rational maps between lifts). Now apply
Steps 1-3 to pairs of curves.
- For a pair S_i, S_j, fix lifts X_i, X_j in H_2 coming from
S_i, S_j respectively.
- If g_0 in G_R carries X_i onto X_j (so g_0 is a real
isometry of H_2 mapping the lift of S_i to the lift of S_j),
then the set of all real such maps is g_0 H_{X_i}.
- By (2.3.a) for i, this set has finitely many classes modulo
right multiplication by Γ_{X_i}.
- Each class containing a rational g admits a rational
representative g (rationality of an isometry on a
Q-structure is preserved by multiplying by the finite-index
integral subgroup Γ_{X_i}).
- Integral changes of lifts add no classes at coarse level
(all lifts are conjugate under Sp_4(Z), which is a finite
union of cosets of Γ); at fine level, only finitely many
resulting classes.
- So there are finitely many rational positive projective
symplectic similitudes g between lifts of S_i and lifts of
S_j for each pair (i, j). This proves (a).
- A dominating isogeny correspondence determines a local
equality of one lifted curve with another (by Proposition
2.2.1(ii) and holomorphicity), hence is represented by one
of these finitely many maps. So all dominating polarized
isogeny correspondences are contained in a finite union of
closed subvarieties. This proves (b).
- Lift each rational g in (a) to an element of GSp_4(Q) by
choosing an arbitrary lift via (2.2.a); the different lifts
differ by rational scalars. Clear denominators by multiplying
with a positive integer: the multiplier mu becomes a positive
integer m, and every corresponding isogeny has degree m^2.
- The set of possible m is finite (finitely many g's per pair,
finitely many pairs), giving (c).
- Finally, (d): passing to a finer fine level just replaces
S_i by a finite cover, and each further lift gives a subset
of the already-considered maps g in general linear (not
projective) form; the underlying GSp_4(Q) similitude does
not change. So M is not enlarged.
To complete the proof of (c), what is needed is the closure statement:
every geometric point of the closure of a local correspondence
admits an isogeny of multiplier m. This goes as follows.
- Work over the fixed fine moduli space at characteristic 0.
- For fixed m, the collection of subgroups K subset A[m] that
are maximal isotropic of order m^2 forms a finite etale scheme
over the moduli space (because A[m] is finite etale and the
isotropic subgroups of a fixed finite group form a finite set).
- Each such K gives the quotient B = A/K with its principal
polarization characterized by q^* lambda_B = m lambda_A.
- The induced map A_2 x A_2 -> A_2 x A_2 (sending s to (s, B_s))
is finite over the first factor, hence proper (second factor
is separated), and has finite fibres (only finitely many B_s
for a given s). So its image is closed.
- The closure of the local Hecke correspondence lies inside
this closed image, so every geometric point of the closure
has an isogeny of multiplier m. This holds even at
singular coarse points and points with extra automorphisms.
- Taking all representatives g and all finitely many curve
components of their pairwise intersections of the products
S_i x S_j gives the required finite union. QED
Remark 2.3.3. The finiteness of the multiplier set M is an
"arithmetic" statement: each g in GSp_4(Q) is an isogeny
between two particular abelian surfaces, and clearing
denominators makes it integral, at the cost of a bounded
multiplier. The optimal M depends on S_i, S_j (via their
embedded moduli) but not on the target points being counted.
Remark 2.3.4. Proposition 2.3.2 is where use is made of the
"fine-level" hypothesis: without a fine level, the only object
dominating S_i would be the moduli space itself, and the
stabilizer H_X might not be discrete.
Needed is the previous finiteness applied to not just one but
all absolute Galois conjugates of the original curve C.
Lemma 2.4.1 (Galois conjugates are Hodge generic).
Let C be a Hodge-generic curve in A_2 over Qbar. Let
sigma in Gal(Qbar/Q). Then sigma(C) is again Hodge generic.
Proof. The canonical (Baily-Borel) model of A_2 has reflex
field Q [Deligne1979, Momford]. Hence Gal(Qbar/Q) acts on
A_2(Qbar) and permutes its special subvarieties
[Orr2021, Corollary 5.3, arXiv version 2]. If sigma(C) were
contained in a proper special subvariety Z of A_2, then applying
sigma^{-1} would place C in the conjugate sigma^{-1}(Z), which
is again a proper special subvariety. Contradicts C Hodge
generic. QED
Let C_1, ..., C_n be the finitely many absolute Galois conjugates
of C. Each is Hodge generic by Lemma 2.4.1. Select fine-level
curves S_1, ..., S_n as in 1.1.1.
Apply Proposition 2.3.2 to this finite list. We get:
a finite set M = M(C_1, ..., C_n) of positive integers,
fixed independent of any target point. Enlarging K for the
centre and the markings adds no new coarse image to this finite
list, so the same M works after any further fixed field
enlargement.
The set M will be used in Sections 4 and 5.
The main result of this section (Theorem 3.2.1) is a purely
analytic-algebraic statement. It takes a target point z_* in a
fixed affine locus, together with:
- a smallness sum (many local places at which one coordinate is
small);
- a collection of analytic relations E_nu vanishing at z_*, with
controlled coefficient sizes;
- a "non-identity" requirement on every algebraic branch of
every containing variety;
and it produces a polynomial bound on the "height parameter" h in
terms of the "complexity parameter" d. The two parameters come from
Section 4: d = [K(t):K] is the degree of the target over K, and h
is the logarithmic height.
The section is technical. Separate the statements ("what the
theorem says" and its lemmas) from the proof ("how this gives there").
A first-time reader can skim the proof of Theorem 3.2.1 and focus
on the top statements; Sections 4 and 5 will show how those
statements are applied.
Conventions. Throughout, h and d are real numbers >= 2. "Poly"
means a fixed polynomial in d with nonnegative rational coefficients;
when the polynomial is called Q_i we mean it to be one of a fixed
finite list. A "controlled" bound in a degree D is one bounded
above by Q(D) times (1 + the joint input height), with the
dependence on height linear, and the ambient dimension fixed.
These notions are made precise in 3.4.
Fix a number field F. For v a place of F, let |.|_v be the usual
(local) absolute value, with |p|_v = p^{-[F_v:Q_p] v_p(p)} for
finite v and the usual, unsquared value for complex places. The
weight of v is
w_v = [F_v : Q_v] / [F : Q]. (3.1.a)
So sum_v w_v = 1 (product formula). For a tuple z = (z_1, ..., z_n)
in F^n, define
||z||_v = max_i |z_i|_v, (3.1.b)
h(z) = sum_v w_v log max(1, ||z||_v). (3.1.c)
This is the standard absolute logarithmic height.
Polynomial evaluation. For a polynomial G in n variables, define
its height to be the height of the full coefficient tuple, with
the additional coordinate 1 included (to handle constant term):
h(G) = h(coeffs of G, together with the constant 1). (3.1.d)
Then for any tuple z, and for each place v,
log |G(z)|_v <= log max(1, ||coeffs G||_v)
+ deg(G) * log max(1, ||z||_v)
+ eps_v * log(#mon(G)), (3.1.e)
where eps_v = 1 at infinite places and eps_v = 0 at finite places.
Interpretation. At finite places, |.|_v is ultrametric; |G(z)|_v
is bounded by max of |coefficient| x |z|^deg, without a binomial
factor. At complex places, the evaluation involves multiplying
coefficients against monomials, giving factor up to #mon(G) which
is bounded by (n + 1)^deg. Summing (3.1.e) over v recovers h(G(z))
as O(deg * h(z) + h(G) + log #mon(G)). The important point is
that #mon does not enter the finite places, so there is no "sum
over all primes" cost.
Embeddings vs places. Sometimes we want to distinguish different
embeddings F -> C or F -> Q_p that induce the same place. In that
case we replace v by an embedding rho with weight
1 / [F : Q]. Then sum over rho equals sum over v (each place v
contributes [F_v : Q_v] / [F:Q] as the total of the weights of its
extensions).
Extension of scalars. If we replace F by a larger number field F',
every tuple z over F is also a tuple over F'. When choose a set
of embeddings of F into C or Q_p-bar, we extend it to F' by taking
ALL extensions of each selected embedding. Then the weights of the
extensions add up to the old weight. So all weighted sums below are
unchanged by extension of coefficients. This is important: we
will freely replace F by larger fields, and the smallness sums
S_small will not change.
One more preliminary notion is needed: "allowed analytic
expression".
Fixed analytic data. Fix:
- a finite list of smooth algebraic curves together with
algebraic connection systems on each;
- finitely many ordinary points ("centres") of these systems;
- rational frames at the centres, regular there;
- algebraic chart germs (inverse coordinate germs) at each centre.
Also fix a total polynomial-combination degree D_0 (e.g., D_0 = 100).
An allowed analytic expression, in the coordinates (a, U) with
a in C and U in C^n, is:
E(a, U) = sum_{|alpha| + |beta| <= D_0}
c_{alpha, beta} a^alpha U^beta
+ sum of fixed-degree polynomial combinations of
the chart germs and horizontal solution matrices,
all coefficient-controlled
Concretely: you're taking a polynomial in (a, U), evaluating the
fixed chart germs at U, taking horizontal solutions Y(a) of the
fixed connection systems at a (and at possibly a second argument
aR if R is one of the coordinates), and combining all of these
with fixed-degree polynomials in the entries.
Only substitution into the fixed germs produces Taylor coefficients
of growing degree; a itself appears polynomially with degree
bounded by D_0.
THEOREM 3.2.1 (Common-relation interpolation).
Fix the analytic data above, an ambient affine space with
coordinates (a, U), and an algebraic ambient locus X in that space.
Let h, d >= 2, let
Psi(h) = 1 + (log h)^c
for a fixed constant c >= 1, and let z_* = (a_*, U_*) in X(Qbar)
with a_* != 0. Suppose the following bounds hold, with fixed
polynomials Q_i (each with nonnegative integer coefficients, in d).
(i) The algebraic locus X has equations of degree at most
Q_0(d) and affine height at most Q_0(d) Psi(h), and the
target satisfies h(z_*) <= Q_0(d) h.
(ii) There is a finite weighted selection V of places (or
embeddings) such that |a_*|_v < 1 for every v in V, and
S_small := sum_{v in V} w_v log(1/|a_*|_v)
>= h / Q_1(d), (3.2.a)
sum_{v in V} w_v log max(1, ||U_*||_v)
<= Q_2(d) Psi(h). (3.2.b)
If a second argument aR is present, we require
|R_*|_v = 1 at every v in V.
(iii) A common list of fixed length s of allowed analytic
expressions
E_1(a, U), ..., E_s(a, U),
each developing as
E_nu(a, U) = sum_{j >= 0} a^j e_{nu, j}(U), (3.2.c)
vanishes at z_* at every v in V. The coefficients satisfy
deg e_{nu, j} <= C (1 + j), (3.2.d)
h(e_{nu, j})_{{nu, 0 <= j <= l}} <= Q_3(l, d) Psi(h). (3.2.e)
There are b_v >= 0 with sum_v w_v b_v <= Q_4(d) Psi(h)
such that the convergent tails at the target, for every
l >= 0 and every v in V, satisfy
|sum_{j >= l} a_*^j e_{nu, j}(U_*)|_v
<= |a_*|_v^l exp((1 + l) b_v). (3.2.f)
(iv) (Non-identity.) If W subset X is an irreducible algebraic
variety containing z_* on which a is nonconstant, then on
every normalization branch above the generic point of
every irreducible component of W cap V(a), the expressions
E_1, ..., E_s are not all identically zero.
Then
h <= C_1 d^{C_2} (3.2.g)
for positive constants C_1, C_2 depending only on the fixed data
and the polynomials Q_i.
Remarks on the theorem.
* Hypothesis (ii) gives "smallness at many places". It is the
only place where the arithmetic inequality |a_*|_v < 1
matters; without it the theorem is vacuous (any target with
a_* = 0 trivially fails).
* Hypothesis (iii) gives "common equations" --- the relations
E_1, ..., E_s must vanish at the target at every selected
place, but are allowed to be a fixed functional combination
of local information.
* Hypothesis (iv) is the mechanism that prevents the descent
from stalling: it ensures that the relations E_nu are not
identically zero on the branches the descent reaches.
* The region V(a) = {a = 0} is the "boundary" of the graph
we construct below; (iv) is about the behaviour at this
boundary. It is an "ordinary-point" condition: the
connection systems are regular at their centres, and the
"non-identity" is at those ordinary points.
The proof takes the rest of Section 3. It is a dimension descent:
we show that z_* cannot have "too large" height without forcing a
polynomial vanishing which is then used (via product formula) to
bound further. Organize the proof as:
3.4 Effective algebra: controlled equations, heights, and
elimination over a fixed large dimension. Gives the
"input/output" bookkeeping.
3.5 Ordinary-point zero estimate: a purely local, purely
algebraic bound on orders of vanishing along a
particular branch, in terms of the branch degree.
3.6 The graph and its special fibre: shows that imposing a
"correction" equation q + g = 0 pulls the graph of
q = f / a^m down by dimension one.
3.7 Layer-by-layer polynomial construction: produces the
polynomial P of bounded degree, controlled height, and
nonvanishing on the graph.
3.8 Assemble: product formula, descent, and end.
The mechanism preview. Suppose h is larger than a polynomial in d.
Descent proceeds through irreducible subvarieties W subset X
containing z_*. At each step:
Step D1: If a is constant on W, the height bound follows
directly from h(a_*) <= Q(d) Psi(h) and S_small.
Step D2: If a is nonconstant, we first form a graph W' of
q = f / a^m for some polynomial f obtained from the
E_nu (of "order" chosen via the zero estimate of
3.5). By the graph lemma 3.6, after imposing one
further equation q + g = 0 on W', this gives a variety
of dimension at most k - 2 (where k = dim W).
Step D3: Using layer-by-layer polynomial construction (3.7),
we find P with P not in I(W'), P(z_*') = 0, and
degree of P bounded. Then W'_new = W' cap {P = 0}
and we pass to one irreducible component containing
z_*.
Step D4: Dimension drops. Repeat.
The key quantitative point: at each step, either h is bounded by
a polynomial in d (in which case we're done), or the equations in
(i)-(iv) constrain the counterexample to a strictly smaller
dimension.
Let us now prove the theorem, filling in 3.3-3.8.
Conventions for the rest of the proof.
Theorem 3.2.1 is the engine that supplies the height bound of
Theorem 5.5.1. Sections 3.4-3.8 prove it. Here is what each
sub-argument contributes.
3.4: Effective algebra in fixed dimension. Grobner basis and
ideal-membership bounds, plus Chow height, give uniform
control: intersections, closures of coordinate projections,
and irreducible components of polynomial-controlled varieties
have degrees polynomial in the input degree and joint affine
heights polynomial in the input degree times the logarithm of
the input height. The ambient dimension is fixed, and all
constants depend on it.
3.5: Ordinary-point zero estimate. Vanishing orders of analytic
derivatives at ordinary points of the connection are bounded
by the algebraic degree of the branch, uniformly in coefficient
heights (Section 3.5). This is the only place the local
differential equations enter.
3.6: The graph and its special fibre. A dimension-reducing
operation on an algebraic variety W containing the target:
replace q = 0 by an arithmetic correction equation
q + (a-derived correction) = 0 and pass to a graph W' inside
W. The special fibre W' cap {a = 0} cuts down dimension by
one (in a controlled way).
3.7: Interpolation by successive layers. The core engine: a
polynomial P of bounded degree, nonvanishing on the graph of
the target, vanishing at the special fibre. Built in layers
via the cohomological remainder argument.
3.8: Assembly. Product formula, descent, and iteration combine
the pieces and yield Theorem 3.2.1.
Fixed notation used throughout: N is the ambient dimension (fixed
small integer). D is the degree. H is the joint affine height
(all H and D bounds below are relative to the input data of
Theorem 3.2.1). C, C_1, C_2, ... are positive constants depending
only on the fixed data.
The proof of Theorem 3.2.1 repeatedly needs to:
- take a variety W subset C^N defined by polynomials of bounded
degree and height;
- form intersections, take closures of projections;
- choose irreducible components;
- test polynomial membership in the ideal of W.
We need all of this to stay "controlled" as defined at the start of
Section 3: i.e., degrees polynomial in the input degree, affine
heights linear in 1 + log(height of input), with the dependence on
the ambient dimension being fixed.
The required facts are all in the literature; we collect them here
so that later steps can invoke them without re-deriving.
Convention. "Input" means a finite list of polynomials
G_1, ..., G_t in C[x_1, ..., x_N], of degree at most D and joint
affine height (with the constant 1 included) at most H.
Lemma 3.4.1 (Grobner basis degree bound).
For any fixed number of variables and any admissible monomial
order, if I is an ideal generated by polynomials of degree at most
D, then I has a Grobner basis of degrees at most Poly(D), with
Poly depending only on N.
Reference: [Dube1990, Corollary 8.3].
Lemma 3.4.2 (Ideal membership representation).
If P is in the ideal I generated by G_1, ..., G_t, with
deg G_i <= D and deg P <= T, then P can be written
P = sum_i A_i G_i,
with deg A_i <= Poly(D, T).
Reference: [Aschenbrenner2004, Theorem 3.4].
Lemma 3.4.3 (Effective height bounds).
Let G_1, ..., G_t be of degree <= D and joint affine height <= H
(the height of the tuple (G_1, ..., G_t, 1) with all coefficients).
Then:
(a) Any such I has a Grobner basis H_1, ..., H_s of degree
<= Poly(D) and affine height <= Poly(D)(1+H).
(b) For P of degree <= T with P in I, the representation
P = sum A_i G_i can be chosen so that the tuple
(A_1, ..., A_s, P) has affine height <= Poly(D, T)(1+H).
Proof. (a) The Buchberger algorithm on polynomial inputs produces
each S-polynomial as a linear combination of the input generators.
Number of steps is bounded by the number of monomials of degree
<= Poly(D), hence polynomial in D. Each step's output has
exponentially inflated height if we don't normalize. But it is possible to
work projectively (in the projective space of coefficient tuples),
normalize each H_i to have exactly one coefficient equal to 1 (or
at least to have its projective norm = 1 fixed), and the resulting
projective Grobner basis has projective heights polynomial in D
times (1 + H), by the elementary fact that "determinant of size s"
has height <= s H + s log s. Dehomogenizing (removing the "1"
coordinate's influence on height) gives affine height <= Poly(D)(1
+ H). (b) ideal membership via Gröbner reduction: express P by
descending through leading terms. At each division step, quotient
gains one monomial factor vs the remainder, i.e. one degree + one
term. Number of steps <= Poly(D, T). Solve the linear system on
coefficients by taking nonzero minors; each minor bounds a
coefficient, and the size of the system is Poly(D, T). So the
height of (A_1, ..., A_s, P) is <= Poly(D, T)(1 + H). QED
For varieties, what is needed is set-theoretic equations rather than full
ideal generators. The relevant tools:
Lemma 3.4.4 (Intersections, projections, components).
Let Z_1, ..., Z_m be subvarieties of C^N with degrees at most D
and joint affine heights at most H. Then:
(a) The intersection Z_1 cap ... cap Z_m has equations of
degree at most Poly(D) and affine height at most
Poly(D)(1+H).
(b) The closure of a coordinate projection of Z_1 has equations
of degree at most Poly(D) and affine height at most
Poly(D)(1+H).
(c) Any irreducible component Z of such a variety admits
equations of degree at most Poly(D) and affine height at
most Poly(D)(1+H).
References: arithmetic Bezout and Chow heights in (Krick, Pardo and Sombra). The
intersection bound uses Corollary 2.11 of (Krick, Pardo and Sombra). The projection
bound uses Lemma 2.6 of (Krick, Pardo and Sombra). Additivity and nonnegativity
of Chow heights are in [KPS2001, Section 1.2.4]. Comparison
between Chow coefficient height and cycle height is [KPS2001,
Lemma 1.1 and inequality (1.2)].
The following construction from Chow form to actual equations will
be used: if Z subset P^N is irreducible of dimension r with Chow
form C_Z, substitute
sum_{j=1}^N T_{ij} (X_j - x_j X_0) = 0, 0 <= i <= r, (3.4.a)
and take coefficients as a polynomial in T_{ij}. The coefficients
vanish precisely at points x with [1:x] in Z. There are finitely
many such coefficients (polynomial in deg Z), each of degree at
most (r + 1) deg Z, of controlled affine height (from the standard
expansion of the Chow form). So a bounded set of generators.
Consequence: every variety W appearing in the proof of Theorem
3.2.1 admits an ideal b of generators with
V(b) = W, b subset of I(W), (3.4.b)
all of the "controlled" size above. We do not need generators of
the radical I(W); membership in b suffices for evaluation on W,
and nonvanishing on W is imposed separately (Section 3.7).
The following lemma is the basic analytic ingredient. It bounds
how fast an allowed analytic expression can vanish along a branch
of a variety W, in terms of the "order of a" on that same branch
and the degree of W. It is a version of the classical multiplicity
estimate of Bertrand-Beukers [BertrandBeukers1985, Theorem 1],
adapted to (i) varying ambient varieties W, (ii) a second argument
aR, and (iii) the need for the bound to be polynomial in the degree
of W.
Lemma 3.5.1 (Uniform ordinary-point order bound).
For the fixed analytic data of Theorem 3.2.1, there is a polynomial
M(D) with the following property.
Let W subset C^N be a reduced irreducible affine variety of degree
at most D on which the coordinate a is nonconstant. Let p be a
germ-point of the normalization of V_W(a) (where V_W(a) is a
codimension-1 subvariety of W). Then every allowed analytic
expression E which is not identically zero on the normalization
near p satisfies:
ord_p E < M(D) * ord_p a. (3.5.a)
The bound is independent of heights of coefficients of W and of
the expression.
The rest of 3.5 is the proof.
Reduction to a curve of controlled degree.
Let k = dim W >= 1. Denote by D~ the normalization divisor defining
the branch (i.e., the divisor in the normalization of W above which
p sits). At a general point p' of D~:
- both the normalization of W and D~ are smooth (after dense open
restriction, in characteristic zero);
- the finite map from D~ to its image D_0 subset W is etale
(separable in characteristic zero).
Choose k - 1 general affine linear functions f_1, ..., f_{k-1}
whose differentials restrict at p' to a basis of the cotangent
space of D_0 at the image of p'. Then the pullbacks f_1, ...,
f_{k-1} restrict to a basis of the cotangent space of D~ at p'.
Consider the common level set C' = {f_1 = f_1(p'), ..., f_{k-1} =
f_{k-1}(p')}. It is a smooth curve germ through p', transverse
to D~ in the normalization. Its image in W is a curve Y through
the image of p' in W.
Two properties:
(i) Y is the level set of k - 1 affine linear functions inside
W, so its projective closure in P^N has degree <= deg W by
Bezout.
(ii) The natural map from the normalization of Y to the
normalization of its slice in the normalization of W is
generically an isomorphism (transversality ensures
birationality, smoothness gives actual isomorphism).
Orders preserved along Y. If ord_{D~} a = e > 0 and
ord_{D~} E = N for the leading expression E in its a-expansion,
only the terms with j <= N/e contribute to the leading coefficient.
After deleting a proper closed subset of D~ it is possible to write
a = t^e v, trunc(f) = t^N u, (3.5.b)
where t is a local equation of D~ and u, v are units. On the
transverse slice Y, t is a uniformizer. Hence the orders on Y
are exactly e and N. All these conditions hold at a general slice
point.
Curve degree and genus. Let T_0 be the smooth projective
normalization of Y. Then deg(Y) <= deg W <= D, so T_0 has degree
<= D as a curve in P^N. By a general birational plane projection,
genus(T_0) <= Poly(D).
Every ambient coordinate on T_0 is a quotient X_i / X_0, so its
polar divisor has degree <= D. In particular, the maps defined by
the arguments a and aR have degrees at most D and 2D.
Cover with fixed maps. Adjoin the points on all fixed curves
specified by the algebraic chart germs. Only the nonconstant
argument maps matter; the constant ones contribute constant chart
factors. For each fixed map phi_i: T_0 -> P^1 of degree d_i and
branch set B_i (branch points plus possibly infinity), form the
normalized fibre product T -> T_0 of all phi_i. Choose one
dominating component.
Then:
- degree of T -> T_0 is at most N_0 = prod_i d_i (fixed).
- T -> T_0 is etale outside union_i phi_i^{-1}(B_i), a set of at
most sum_i #B_i * deg(phi_i) = O(D) points.
- ramification contribution above each branch point <= N_0.
- So Riemann-Hurwitz gives
2 g(T) - 2 <= N_0 max(0, 2 g(T_0) - 2) + O(D). (3.5.c)
- The degrees of pulled-back fixed divisors (from all fixed
connections) are bounded by N_0 deg(phi_i).
- All ramification indices are bounded. Any further ramification
multiplies the two orders in (3.5.a) by the same factor.
So now we are on a fixed curve T with controlled genus and
controlled degree maps to the fixed curves, and we still have
ord_p(phi(a)) = e, ord_p(phi(E)) = N
at the point p above the original branch. We need to bound N.
The differential system. On T, the expression E lifts to
E = r_0 y, (3.5.d)
where y is a horizontal column of the fixed-rank system and r_0 is
a rational row. Details:
- In direct sums and tensor products of the fixed pulled-back
systems, all polynomial algebraic factors (from substituting
into the chart germs) are absorbed into r_0.
- The rational frames are the pullbacks of the fixed frames;
they are regular and invertible at p, since phi_i is etale at
p for the constant components, and for a and aR we are at the
centres (a = 0 there, R is regular there).
- The connection and r_0 are regular at p. Their global polar
divisors have degrees polynomial in D, by the degree bounds on
the pullback maps and the coordinate functions.
Uniformizer. Since T is a smooth projective curve, choose a
rational function z on T which is a uniformizer at p, and whose
polar divisor has degree at most 2g(T) + 1. Existence by
Riemann-Roch: prescribe first two jets at p, allow a pole at
another point. (In the case g(T) = 0, a degree-1 rational function
suffices.)
Logarithmic derivative. Put delta = d/dz. Write the horizontal
equation as
delta y = A y, (3.5.e)
where A is a rational End-matrix (of the fixed rank). The poles of
the rational vector field delta are bounded by the zero divisor of
dz, hence by
deg(zero dz) <= 2 g(T) - 2 + 2 deg z = O(D^2).
Differentiating a rational function increases its reduced polar
divisor by at most its pole divisor, plus the vector field's pole
divisor. Hence A, r_0, and any fixed number of their derivatives
have polar degrees polynomial in D. They remain regular at p,
using that z is a uniformizer.
Successive rows.
r_{i+1} = delta r_i + r_i A. (3.5.f)
Then
delta^i E = r_i y. (3.5.g)
The successive rows r_0, r_1, ... become dependent after at most N
steps (N = rank of the fixed system). Let rho be the first index
where r_rho is in the C(T)-span of r_0, ..., r_{rho-1}.
M := span_{C(T)} (r_0, ..., r_{rho-1}). (3.5.h)
Differentiating the dependence shows that r_{rho+1}, ... all lie
in M (because the same structure of the connection produces the
next derivative of a linear combination in M). So M is stable
under the dual connection.
The quotient. Let O = O_{T, p} be the local ring at p. Saturate
the common kernel of the rows of M in O^N:
K := { v in O^N : r_i v = 0 (1 <= i <= rho) } localized. (3.5.i)
(In practice, saturate by allowing denominators away from the
exceptional poles.) Saturation makes K a subbundle locally. K is
stable under the ordinary connection delta, since its derivative is
regular and belongs to K generically.
The quotient Q = O^N / K has an induced ordinary connection.
Concretely: if v is a section of K, then delta v has regular
coordinates at p and satisfies r (delta v) = 0 for all r in M (by
stability of M under the dual connection). So delta v is in K.
Consequently the induced connection on Q is regular (no pole).
The image y-bar of y in Q is a nonzero horizontal section. If
y-bar = 0, then E = r_0 y = 0 (since y is in K, which is annihilated
by r_0). Conversely, if E identically zero, then all delta^i E =
r_i y vanish, so y is in K, so y-bar = 0. So y-bar nonzero
<=> E not identically zero.
Coefficient recurrence. The ordinary linear differential equation
on Q (regular, no pole) forces the following: if y-bar has zero
reduction at p, then y-bar is identically zero. Specifically, the
recurrence for Taylor coefficients near p is: leading coefficient
of y-bar at order 0 vanishes, then by the ordinary differential
equation all subsequent coefficients vanish; contradiction.
So y-bar has nonzero reduction at p.
Saturated row lattice. The integral dual of the free (locally)
quotient Q is exactly the saturated row lattice
L := M intersection O^N. (3.5.j)
Its rows are the integral functionals annihilating the common
kernel. Since y-bar has nonzero reduction at p, some functional
lambda in L has (lambda y) a unit in O.
Elementary divisors. Choose a nonzero rho x rho minor Delta of the
matrix with rows r_0, ..., r_{rho-1}. All these rows are regular
at p (only finitely many algebraic poles, and we work near p).
Cramer's rule on the selected columns yields
Delta * lambda = sum_{i=0}^{rho-1} c_i r_i, (3.5.k)
with c_i in O. Applying to y:
Delta * (lambda y) = sum_i c_i (r_i y).
Since lambda y is a unit, ord_p(Delta lambda y) = ord_p(Delta).
Some index i must satisfy
ord_p(r_i y) <= ord_p(Delta). (3.5.l)
Order bound. A nonzero rational function on T has equal total
zero and pole degrees. So ord_p Delta <= (its total pole degree),
which is bounded by the polar degree of Delta, which is Poly(D).
Finally, since z is a uniformizer at p and delta = d/dz,
ord_p(E) = ord_p(r_0 y) <= ord_p(delta^i E) + i = ord_p(r_i y) + i <= ord_p(Delta) + N - 1.
So ord_p(E) <= Poly(D) (with an additive N-1 and Delta-pole-degree
correction). Increasing the resulting polynomial M(D) by 1
proves (3.5.a), because ord_p a >= 1.
Important: the whole argument used only:
- degrees / genera / ramification / fixed ranks
- the fact that the connection is ordinary at the centre
- the Bertrand-Beukers classical framework
and did NOT use heights of the coefficients of W or of the
expression E. This is what makes the bound uniform.
QED
We now set up the auxiliary graph used in the descent. The lemma
below is a purely algebraic statement. Its role in the descent will
be to make the special fibre of the graph misbehave in a specific
way: the graph's intersection with the divisor a = 0 must have
dimension one less than its special fibre boundary, once we impose
one further equation.
Lemma 3.6.1 (Constrained graph fibre dimension).
Let W subset C^N be an irreducible affine variety of dimension
k >= 1 with coordinate a nonconstant. Let f be a regular function
on W, and let m >= 1 be an integer. Suppose
ord_p f < m * ord_p a (3.6.a)
for every normalization branch p over the generic point of
V_W(a). Let W' subset C^{N+1} be the affine graph closure of
q = f / a^m (3.6.b)
over W \ V(a). Then for every regular function g on W:
dim V( I(W'), a, q + g ) <= k - 2. (3.6.c)
When k = 1, the set is empty; the same convention applies if
V_W(a) is empty.
Proof. Write A = C[W], u = f/a^m, and let B be the finite
normalization of A. The graph has coordinate ring A[u]; B[u] is
finite over it. Suppose there were a point of the graph lying
above a generic point p of a component of V_W(a). Lying-over in
B[u] gives a prime r above p, i.e., r cap B = p-cap-B. Its
localization B_r is a discrete valuation ring.
Hypothesis (3.6.a) means: in B_r, f = pi^N * v with N < me, and
a = pi^e * v', with pi a uniformizer, v and v' units. Then
u = f/a^m = pi^{N - me} * (v/(v')^m), with N - me < 0. So
u = pi^{-s} * w where s = me - N > 0 and w is a unit. In
particular u has a pole at r.
The map from the graph to B[u]-locus would send (r, u) to some
point (r, u_r). In the local ring of this point, pi^s u_r = w
where s > 0 and pi is in the maximal ideal. But u_r is an element
of the local ring of an affine scheme, w is a unit, and pi^s is
in the maximal ideal's s-th power. The equality pi^s u_r = w
would require pi^s to divide a unit, contradiction.
So the constructible image of V_{W'}(a) under the projection to W
contains no generic point of V_W(a). Each component of the latter
has dimension k - 1. Hence the closure of this image has dimension
at most k - 2.
Now impose the further equation q = -g. On the graph W', the
projection (a, U, q) -> (a, U) is a closed immersion on the
intersection with {q = -g}: because q is already determined as
-g(a, U), the extra coordinate is redundant. So the full
intersection V(I(W'), a, q + g) maps injectively with image inside
the already-shown-constructible image of V_{W'}(a), which has
dimension at most k - 2. Therefore (3.6.c) holds. QED
Why the extra equation q + g = 0 matters. Here is a simple
example. Let W = C^2 with coordinates (a, u), f = u, m = 1. The
graph W' = {a q = u} is a smooth surface in C^3, with coordinate
projection (a, u, q) -> (a, u). Over the point (a, u) = (0, 0),
the "fibre" of the projection is the entire q-line, because
0 * q = 0 for all q. So the special fibre of the graph alone is
1-dimensional (a line) over the 0-dimensional image point.
However, once we impose a further equation q + g_0(u) = 0, the
q-line is cut down to a single point: q = -g_0(0). So the
constrained graph fibre is 0-dimensional over the 0-dimensional
image, giving k - 2 = 0.
The figure below sketches the situation. Horizontal arrows are
projections, vertical arrows are inclusions.
+--------------------+ +--------------------+
| W' = {u = a q} | projection | W = A^2 |
| coord (a, q) | ------------> | coord (a, u) |
+--------------------+ +--------------------+
^ ^
| |
| |
+--------------------+ +--------------------+
| W' cap {a = 0} | contracts | image {(a,u) = 0} |
| q-line, dim 1 | ------------> | dim 0 |
+--------------------+ +--------------------+
^
| q + g_0(u) = 0
|
+--------------------+
| a = 0, q = -g_0(0) |
| dim 0 |
+--------------------+
The graph boundary is a line, but its image is a point. The
equation q + g_0(u) = 0 selects one point on that line, giving the
dimension k - 2 = 0 that the layer count uses.
The next lemma is the technical engine of the descent. It produces
a polynomial P of bounded degree that vanishes at the (lifted)
target, is not in the ideal of the graph, and is a specific
combination of the graph's defining equations, one additional
polynomial F_n of controlled degree, and a high power a^n of the
parameter. The form of the combination is what makes the local
evaluation of P(z_*') controlled.
Lemma 3.7.1 (Interpolation through successive layers).
Let W' subset C^M be irreducible of dimension k >= 1, with
coordinates including a, U, q. Let b = (G_1, ..., G_t) be generators
with V(b) = W' and controlled degrees and heights. Suppose
dim V(b, a, F_0) <= k - 2, (3.7.a)
where F_0 = q + g_0(U). Put, for n >= 1,
F_n = F_0 + sum_{j=1}^{n-1} a^j g_j(U), (3.7.b)
where m >= 1 is an integer and deg g_j <= C(m + j + 1). Here M
and C are fixed, and all complexities including m are polynomially
bounded in d; the coefficient heights through index n are bounded
by Q(n, d) Psi(h).
For any prescribed positive integer r that is polynomially bounded
in d, there are positive integers L and n = r L, both polynomially
bounded in d, and a polynomial P such that:
deg P <= L, (3.7.c)
P not in I(W'), (3.7.d)
P = sum_{lambda=1}^t A_lambda G_lambda + A F_n + a^n B. (3.7.e)
The degrees of all the coefficients in this representation are
polynomial in d, and their joint affine height, including that of
P, is at most Q'(d) Psi(h). The polynomial Q' may depend on the
prescribed polynomial bound for r.
Proof.
Step 1. Grobner basis of j = (b, a, F_0). Fix a degree-compatible
monomial order. Lemma 3.4.3 gives a Grobner basis of j, together
with represenations in the displayed generators, of controlled
degrees and heights.
Step 2. Division identity. For a polynomial H of degree at most T,
Grobner division gives
H = rem_j H + sum_lambda C_lambda G_lambda + a C + F_0 D, (3.7.f)
with the right side all of degree at most T + Delta, where Delta is
a fixed polynomial in the input complexities independent of T. The
point: the quotients C_lambda have degrees <= T - deg(leading) and
their multiplication by the Grobner elements adds at most Delta in
degree, but does not scale with T.
Step 3. Work modulo (b, F_n, a^n). Modulo these, (3.7.f) becomes a
formal power-series identity in a, of length n in a-power expansion:
H == (rem_j H) + a C + F_0 D (mod j)
But F_0 D = (F_n - sum_{j=1}^{n-1} a^j g_j) D - F_n D = ...
More usefully: modulo (b, F_n, a^n),
F_0 D == -sum_{j=1}^{n-1} a^j g_j D (mod F_n, mod a^n),
which advances the a-adic valuation by at least one. Similarly the
a C term advances by one. So each application of (3.7.f) to a
layer-index-i coefficient drives the result to a strictly higher
layer index.
Step 4. Layer decomposition. Start with any polynomial H_0 of
degree <= L. Apply the reduction of (3.7.f) to the coefficient of
1 (layer 0), of a (layer 1), ..., of a^{n-1} (layer n-1). At
layer i, the coefficient being divided has degree at most
L + B_0 * (i + 1),
for a polynomially bounded B_0 (via the substitution of g_j of
degree <= C(m + j + 1) and the Delta of 3.4). There is no
substitution at the same layer, so the layer index strictly
increases at each step.
Step 5. Conditions. Retain every remainder rem_j H at every layer.
Require it to be zero. This gives a linear system in the
coefficients of H_0.
When k = 1, the hypothesis (3.7.a) says the ideal j is the unit
ideal (V(j) = empty), so there are no remainder conditions at all.
Otherwise k >= 2. The number of standard monomials of j, of degree
at most T, is at most
B_1 * (1 + T)^{k-2},
with B_1 polynomially bounded in the input complexities. This is
the elementary bound: if the monomial initial ideal has generators
of degree at most D_1, then in a standard monomial at most
dim V(j) exponents can be >= D_1 (otherwise the support of a
monomial generator would lie in the variables with large exponents),
and every other exponent is bounded by D_1 - 1.
So the number of linear conditions on H_0 is at most
B_1 * n * (1 + L + B_0 n)^{k-2}. (3.7.g)
Step 6. Choosing H_0. Choose k algebraically independent
coordinate functions on W'. Let V_L be the space of their
monomials of total degree at most L. Then
dim V_L = binom(L + k, k), V_L cap I(W') = 0. (3.7.h)
The second identity: these monomials, as polynomials in the
ambient coordinates, are linearly independent modulo I(W')
(because they restrict to algebraically independent functions on
W' of independent monomials).
Set n = r L. For a proper ideal j, (3.7.g) is at most
B_2 * r * (1 + B_0 r)^{k-2} * L^{k-1}
for B_2 polished. For L sufficiently large (polynomial in d given
the bound on r), this is smaller than binom(L + k, k). So there
is a nonzero H_0 in V_L whose remainders all vanish. Setting P =
H_0 gives (3.7.c), and (3.7.e) from the accumulated representation.
P not in I(W') follows from V_L cap I(W') = 0.
Step 7. Height bounds. The coefficient transformations in Grobner
division have controlled affine heights: for matrices A of inner
dimension b, h(AB) <= h(A) + h(B) + log b. So the n layer
operations have polynomial heights. The source V_L is spanned by
monomials (height 0). A kernel vector obtained by minors and
normalized to have one entry equal to 1 has affine height
Q(d, n, L) Psi(h). Since r, L, n are polynomial in d, this is
Q'(d) Psi(h). QED
Remark 3.7.2. The construction "layer by layer" is essential: a
direct single-level interpolation would produce a polynomial of
degree growing with the wrong variable. The n = r L rescaling
gives the ratio n / L any prescribed polynomial in d, so the "effort"
from the layer expansion can be made to dominate the analytic term
in (3.8.e) below.
Remark 3.7.3. What P looks like. The polynomial P is a member of
the small-degree vector space V_L (so of controlled ambient degree),
vanishing at the (lifted) target z_*', and nonvanishing on W'.
Its coefficients are the solution of a linear system, controlled
in height by minors.
We now assemble the pieces and prove Theorem 3.2.1.
Setup. Assume for contradiction that h is larger than a
polynomial in d. We will derive a contradiction.
The descent. We descend through irreducible subvarieties
W_0 supset W_1 supset W_2 supset ...
of X, each containing the target z_*, starting with W_0 = an
irreducible component of X containing z_*. Invariants maintained:
- W_j has bounded equations degree Poly_j(d) and affine height
Q_j(d) Psi(h);
- z_* is in W_j;
- the analytic data, places, and tails of Theorem 3.2.1 apply
verbatim.
Initially these hold by (i) and (ii) of Theorem 3.2.1 (with the
input bounds). We will show that either h is bounded by Poly(d),
or the descent can produce W_{j+1} with dim W_{j+1} < dim W_j and
the invariants preserved.
Case A: a is constant on W_j.
Let a_* = the constant value. Then h(a_*) <= Q_j(d) Psi(h) by
the component-height bound. But the product formula (3.1.c) and
the smallness bound (3.2.a) give
h(a_*) = sum_v w_v log max(1, |a_*|_v^{-1})
>= S_small (3.8.a)
>= h / Q_1(d).
Combining: h / Q_1(d) <= Q_j(d) Psi(h). Iterating
(Psi(h) = 1 + (log h)^c), this gives h <= Poly(d). Contradiction.
Case B: a is nonconstant on W_j and V_W_j(a) is empty.
Then the ideal (b_{W_j}, a) is the unit ideal in C[W_j]. Lemma
3.4.3(b) applied to the identity "1 in (b, a)" gives
1 = sum_i A_i G_i + a B, (3.8.b)
with polynomial degrees in d and joint affine height Q(d) Psi(h).
At the target, 1 = a_* B(z_*). Evaluating B on the selected
embeddings costs only its coefficient height, times the local
sizes |a_*|_v (which are < 1 at the selected places) and of
||U_*||_v (which is Q_2(d) Psi(h) locally). So
S_small = sum_v w_v log |B(z_*)|_v
<= Q(d) Psi(h). (3.8.c)
Again combining with (3.2.a) gives h <= Poly(d).
Case C: a is nonconstant on W_j and V_W_j(a) is nonempty.
Let k = dim W_j >= 1.
Subcase C.1: We first build the graph. By (iv) of Theorem 3.2.1,
the expressions E_1, ..., E_s are not all identically zero on any
branch of V_W_j(a). The zeros of a are at most the degree of W_j,
counted with multiplicities (via a general transverse linear slice;
each transverse curve component of degree at most Poly(d) has
normalization whose zero divisor is at the polar degree of a).
At each branch, choose the lowest nonzero order among the E_nu.
Set t = a positive integer in the combination
E = sum_{nu=1}^s t^{nu-1} E_nu in C[[a]] (phi-form).
Each branch excludes at most s - 1 values of t (the leading-coefficient
polynomial is a low-degree polynomial in t, not identically zero by
(iv)). So a permissible t of polynomial size in d exists.
The combined E preserves the coefficient and tail bounds: at
finite places its coefficients are integral (up to fixed
denominators), and the extra triangle-inequality factor at infinite
places has total logarithm polynomial in d. Absorb that factor in
the b_v.
Write
E = sum_{j >= 0} a^j e_j(U).
By Lemma 3.5.1 (zero estimate), an integer m polynomially bounded
in d can be chosen so that, on every relevant branch p,
ord_p f < m * ord_p a, (3.8.d)
where
f = sum_{j=0}^{m-1} a^j e_j(U).
Indeed: the omitted terms sum_{j >= m} a^j e_j have order at
least m * ord_p a (since the e_j are regular on W_j, the a^j
factor contributes at least m * (ord_p a) at each j >= m), and the
rest is bounded by the zero estimate.
Subcase C.2: Form the graph. Let W_j' subset C^{N+1} be the
affine graph closure of
q = f / a^m
over W_j \ V(a). Its equations with controlled support are
obtained by saturating (b_{W_j}, a^m q - f) by a (in a suitable
big polynomial ring with one extra variable); the closure of its
support is exactly the graph. Lemma 3.4.4 gives controlled
equations.
The target lifts to z_*' = (a_*, U_*, q_*) in W_j', where
q_* = a_*^{-m} * (f(z_*)) = sum_{j >= m} a_*^j e_j(U_*) / a_*^m.
Wait, careful: q_* should be the value defined by the graph, which is
f(z_*)/a_*^m. Since a_* != 0 this is well-defined. Using that
E(z_*) = 0 (from (iii), the expressions vanish at target),
there is sum_{j>=0} a_*^j e_j(U_*) = 0, hence
f(z_*) = sum_{j=0}^{m-1} a_*^j e_j(U_*)
= -sum_{j >= m} a_*^j e_j(U_*).
So
q_* = -sum_{j >= m} a_*^j e_j(U_*) / a_*^m. (3.8.e)
This is the representation used in the local evaluation.
Height of lifted target. Using the rational expression for q_*:
h(z_*') <= Q_5(d) h + Q_6(d) Psi(h). (3.8.f)
Where Q_5 is fixed at this stage, before the interpolation degree
and order are chosen.
Local size of q_*. From the tail bound (3.2.f) applied to E with
the chosen coefficient bounds,
|q_*|_v = |sum_{j >= m} a_*^j e_j(U_*) / a_*^m|_v
<= exp((m + 1) b_v). (3.8.g)
So the sum of the positive local logarithmic sizes of (U_*, q_*)
is at most Q_7(d) Psi(h).
Subcase C.3: Apply layer interpolation. Introduce F_0 = q + g_0(U)
where g_0 = e_m(U), and F_n = F_0 + sum_{j=1}^{n-1} a^j e_{m+j}(U).
The dimension constraint "dim V(I(W'), a, F_0) <= k - 2" holds by
Lemma 3.6.1 applied to W = W_j, f, m chosen in C.1. For a controlled
ideal b of W', V(b, a, F_0) has the same set and the same dimension.
Apply Lemma 3.7.1 with the prescribed ratio r = n / L to be chosen
immediately. Get L, n, P such that (3.7.c) - (3.7.e) hold.
At the lifted target z_*', (3.7.e) becomes
P(z_*') = A(z_*') F_n(z_*') + a_*^n B(z_*'). (3.8.h)
The tail bound for F_n at the target:
|F_n(z_*')|_v <= |a_*|_v^n exp((m + n + 1) b_v). (3.8.i)
Subcase C.4: Product formula. Bound the value of P(z_*') place by
place.
At selected places v in V: (3.8.h), (3.8.i), and the coefficient
bounds of Lemma 3.7.1 give
|P(z_*')|_v <= |a_*|_v^n exp(Q_8(d, n, L) Psi(h)). (3.8.j)
Summing:
sum_{v in V} w_v log |P(z_*')|_v
<= -n S_small + Q_8(d, n, L) Psi(h). (3.8.k)
Here the -n S_small contribution comes from the (|a_*|_v)^n factor
at |a_*|_v < 1. No error proportional to the number of finite places
appears (we used the ultrametric and the tail bound; see 3.1).
At places outside V: (3.8.f) plus the coefficient bounds of Lemma
3.7.1 give, since deg P <= L,
sum_{v notin V} w_v log |P(z_*')|_v
<= L Q_5(d) h + Q_9(d, n, L) Psi(h). (3.8.l)
Case ending. Choose n, L in this order. First choose an integer
r > 2 Q_1(d) (Q_5(d) + 1) + 1, (3.8.m)
polynomially bounded in d. Next choose the polynomially large L
supplied by Lemma 3.7.1, and set n = r L.
Suppose P(z_*') != 0. Its product formula says sum_v w_v log
|P(z_*')|_v = 0, but (3.8.k) + (3.8.l) give
0 <= ( -n / Q_1(d) + L Q_5(d) ) h + Q_10(d, n, L) Psi(h). (3.8.n)
The coefficient of h is at most -L (by the choice (3.8.m)). As
n, L are fixed polynomials in d, this inequality implies
h <= Poly(d). Contradiction. So P(z_*') = 0.
Subcase C.5: Descend. Since P not in I(W_j'), the intersection
W_j' cap {P = 0}
has dimension at most k - 1 (k = dim W_j). Project to the
original (a, U)-space; take its closure; choose an irreducible
component containing z_*.
Such a component exists because a_* != 0 and the graph projection is
an isomorphism over that open set. Projection cannot increase
dimension. Discard the graph coordinate q, obtaining a new
subvariety
W_{j+1} subset W_j
with dim W_{j+1} < dim W_j = k. Equations of bounded degree and
height (via Lemma 3.4.4). The target z_* is in W_{j+1}.
At the next step, construct a new graph from the original common
equations on this smaller variety. The ambient number of variables
stays fixed (dim increases by 1 for the graph coordinate, then drops
when we discard q).
The dimension drops at every continuing step, so there are at most
dim X such steps. A zero-dimensional final W has W = {z_*} and a
constant, and is covered by Case A. Taking the maximum of the
finitely many polynomial thresholds proves the theorem.
QED
We now supply the arithmetic inputs that Theorem 3.2.1 needs. The
main objects:
- integral endomorphisms of A_t (small, controlled);
- de Rham coordinate bounds in terms of the global height;
- potential good reduction at each prime;
- the "pole ideal" control on denominators of endomorphism
coordinates in a fixed rational frame;
- common local relations between centered and target
endomorphisms at selected places;
- a grouping that preserves smallness.
Input (i)-(iv) of Theorem 3.2.1 will be assembled at the end of
this section (Section 4.5).
At a QM point t the 2-plane E_t is spanned by actual algebraic
endomorphisms. Two facts make those endomorphisms controlled: (i)
the Galois action on the geometric endomorphism lattice has bounded
order, so a bounded extension L/K(t) defines them all, and (ii) the
Masser-Wustholz endomorphism estimate yields a spanning set whose
Rosati norm grows only polynomially in d and h. Together these give
the "small labeled integral pair" used below.
Lemma 4.1.2 (Small labeled integral pair).
There are constants c, C > 0 with the following property. For
every target t, some extension L/K(t) of degree at most C defines
all geometric endomorphisms of A_t. There are two independent
endomorphisms u_1, u_2 in End_L(A_t), symmetric and trace free,
spanning E_t, satisfying
q_lambda_t(u_j) := (1/4) Tr(u_j u_j^dagger)
<= C (d h)^c, j = 1, 2. (4.1.a)
Writing P = (P_1, P_2) for the de Rham coordinates of these
endomorphisms in the fixed frame, a labeled target means a target
together with its chosen pair; any field conjugate of a labeled
target is required to conjugate the pair as well. The pair P need
not generate the saturated lattice L_t = E_t cap V_{Z,t}.
Proof.
Field of definition. The Galois action on the geometric
endomorphism lattice is a finite subgroup of GL_r(Z) with r <= 16.
A finite subgroup of GL_r(Z) injects into GL_r(F_3) via reduction
modulo 3 (the kernel of reduction is torsion free), so its order
is bounded purely in terms of r. The fixed field of this kernel
is the required L, and [L : K(t)] <= C.
Masser-Wustholz. For a principally polarized abelian variety of
fixed dimension, the endomorphism estimate of (Masser and Wustholz) produces a
rational spanning set of integral endomorphisms whose Rosati
lengths are bounded by a fixed power of
max(2, [L : Q], h_F(A_t)).
The hypothesis "polynomial dependence on [L:Q]" is explicit in the
Proposition of [MW94, Section 4], with its parameter m equal to the
dimension of A_t; the "smaller image" alternative is then zero.
Its lattice Lemma 2.1 supplies a basis with the same form of bound.
The theta-height comparison [Pazuki12, Theorem 1.1 and Corollary
1.3] at one fixed theta level, together with functoriality of
heights on the fixed curve S, gives h_F(A_t) <= C h. The degree
of passage to that theta level is bounded. Since [L : Q] <= C d,
this gives a Q-spanning endomorphism set whose Rosati lengths are
at most C (d h)^c.
Projection to the 2-plane. The map
u |-> 4(u + u^dagger) - Tr(u + u^dagger) id
sends integral endomorphisms to integral trace-free symmetric
endomorphisms, has rational image of dimension two by Lemma 1.3.1,
and increases Rosati norm by at most a fixed factor. Apply it to
the set from MW and select two independent images; these become
u_1, u_2, satisfying (4.1.a).
Norm and trace invariance. Both q_lambda_t and Tr are traces of
actual endomorphisms and their Rosati adjoints; these are stable
under field conjugacy. Hence (4.1.a) holds for every conjugate of
u_1, u_2. QED
Corollary 4.1.3. After the bounded extension of Lemma 4.1.2, the
de Rham coordinate pair P = (P_1, P_2) in the fixed rational
frame is well defined and satisfies the bounds of Proposition
4.2.1 below.
The pair u_1, u_2 of Lemma 4.1.2 is integral and bounded. We need
to bound the height of its de Rham coordinate pair P = (P_1, P_2)
in the fixed rational frame on S. Only the frame itself may have
denominators moving with t.
The de Rham cohomology H_{dR} = R^1 pi_* Omega^* has an integral
lattice structure coming from a proper integral model of the
abelian scheme, though the lattice depends on the reduction type.
The poles of the rational frame appear when the ambient cohomology
degenerates; we control these by a fixed pole ideal on the smooth
completion.
Setup. Fix the rational frame e_1, ..., e_5 of V_{dR}, regular
and invertible at the fixed center t_0. Let
J = J(ref frame, t_0)
be an ideal on Sbar whose local multiples measure the failure of
the frame to be integral anywhere on S. Its construction and
boundedness are Lemma 4.3.3 below.
Proposition 4.2.1 (Total size of the de Rham coordinates).
With the frame e_1, ..., e_5 as above, for every QM point t in
S(K) (after the bounded extension of Lemma 4.1.2), the coordinate
pair P = (P_1, P_2) of (u_1, u_2) in the frame satisfies
sum_v w_v log^+ ||P_1, P_2||_v <= C (h + log d). (4.2.a)
At any place v of K(t) outside a fixed finite set of primes, the
per-place bound
||P_i||_v <= exp(C (h_v + lambda_{J, v}(t) + eps_v d log d)) (4.2.b)
holds, with eps_v = 0 for finite v. The bound depends on t only
through (d, h, J(t)).
Note. We do NOT assert that the coordinates are integral at any
prime; only the pole ideal J controls the denominators. The
denominators can blow up as t approaches the boundary (J(t) -> +inf),
but only linearly in t's own height.
Proof of (4.2.a) and (4.2.b).
Step 1. Integral lattice. At a prime p of K where A -> S extends
to a good model over a smooth model of S, the de Rham cohomology
sheaf H_{dR} has a natural integral structure. Away from finitely
many fixed primes (unramified in K, p | N, etc.), this integral
structure exists and is compatible with the frame after multiplying
by a bounded rational function.
Step 2. Endomorphism integrality. An endomorphism extends to the
good model and acts on integral de Rham cohomology as an integral
matrix. Therefore its coordinates in an integral Z-basis are
bounded in each p-adic absolute value by 1.
Step 3. Passage to the rational frame. The change-of-basis from
the integral de Rham lattice to the rational frame e_j (defined
over K) has denominators measured by an ideal J_0 (on the smooth
model). If n_p is the valuation of a p-generator of J_0 at p, the
frame entries lie in p^{-n_p} times the integral lattice, and the
dual frame entries in p^{-n_p} times the integral dual lattice.
Endomorphism coordinates in the frame are then bounded by p^{2n_p}.
Step 4. Summation. At finite places, the bound p^{2 n_p} gives a
contribution of order h(J_0(t)) = O(h). At archimedean places,
the Hodge-norm bound
||P||_sigma <= ||v||_H <= q_lambda_t(v)^{1/2} <= C (dh)^{c/2}
gives C (log d + log h) = O(h) + O(log d). Summing all places
with weights gives (4.2.a).
Step 5. Center behaviour. Near t_0, J(t) -> 0 and the
coordinates P_i are small. This is why the "small endomorphisms"
local system is available from t_0, and it enters Section 4.4.
QED
The pole ideal J is constructed in Lemma 4.3.3 below with an
explicit height bound. It depends only on the fixed frame, the
fixed center t_0, and a fixed finite set of primes.
The height h(J(t)) is bounded by O(h(t)) via the standard
"height of an ideal at a point" inequality, so the log J term in
(4.2.b) is bounded by O(h) and gives (4.2.a) uniformly in t.
Two arithmetic facts are needed:
(i) Every QM point has good reduction at every prime, after a
bounded-degree extension of the field of definition. This
is standard potential good reduction for abelian varieties
in a fine moduli space.
(ii) The rational frame e_1, ..., e_5 of V_{dR}, which is
regular at the fixed center t_0, fails to be integral on
S outside a controlled divisor. We package this failure
as a fixed coherent ideal J (the "pole ideal"), such that
integral endomorphism coordinates are bounded by J.
Both are technical but standard. We state them and outline the
proofs.
Lemma 4.3.1 (Potential good reduction).
Fix the fine level N >= 3, and K a number field containing the
reflex field of the fine-level Siegel moduli. There is a constant
C = C(N) such that: for every abelian scheme A -> S over K and every
QM point t in S(K), and for every finite prime p of K, there is a
finite extension K' / K with
[K' : K] <= C (4.3.a)
such that A_{K'} -> S_{K'} has good reduction at every prime p' of
K' above p.
Further: the extension can be chosen uniformly in t, depending only
on the fine level N and the dimension g = 2.
Proof. By the theory of moduli spaces (see [Chai1985, Chapter I,
Theorem 1.4]), the fine moduli space A_2(N) over O_K[1/N] is
smooth and proper (after a smooth resolution and finite cover, if
needed). Every point of A_2(N)(K) extends to a point of
A_2(N)(O_{K'}[1/N]) for a possibly larger field K' by properness
of A_2(N) over O_K[1/N]; distinct extensions correspond to
distinct points in the special fibre of the moduli space, which is
0-dimensional for fixed coarse image. The size of the reduced
moduli space (over the residue field, if p unramified) is a constant
#{A_2(N)(F_{p^e}) : e <= some bound} bounded in terms of N and g.
Alternatively, use [BLR1990, Section 1.2, Proposition 8]: an
abelian variety with a polarization of degree prime to p and a
full level-N structure with (N, p) = 1 acquires a good reduction
after a finite extension of degree bounded by #Sp_4(Z/N) * C(p).
Passing to the compositum over all primes p | N and multiplying
by bounded constants gives (4.3.a).
For p | N, one uses the same technique but over the residue ring at
p, via Barsotti-Tate / Serre-Tate theory of the moduli problem of
level N. In any case the extension degree depends only on N.
QED
Remark 4.3.2. We do not need to actually compute K' or the bound
C. We just need that K' can be taken inside a fixed algebraic
closure with degree bound C, so that passing to K' adds a bounded
factor to h(t) and preserves all the polynomial d-, h-, and Psi(h)-
bounds of Sections 3 and 4.
Lemma 4.3.3 (Pole ideal).
Let e_1, ..., e_5 be a rational frame for V_{dR} on a smooth model
of S extending to Sbar, and let t_0 be the fixed center. Assume
the frame is regular and invertible at t_0 (this is the choice from
Lemma 1.1.1). Then there is a coherent ideal J = J(frame, t_0) on
Sbar, fixed once and for all, such that for every point t of S not
in a fixed finite exceptional set E, and every integral
dagger-self-adjoint trace-zero endomorphism v of A_t (over a bounded
field extension of K(t)) with q-coordinate P, the de Rham coordinates
P_1, ..., P_5 of P in the frame e_1, ..., e_5 satisfy
|P_i|_v <= exp( C_v * ( delta_v, E + lambda_{J, v}(t) ) ) (4.3.b)
at every finite place v of K(t) not in E, for a fixed small function
C_v depending only on the frame.
Further,
sum_v w_v log max(1, ||P||_v) <= C (h(t) + d log d + 1), (4.3.c)
where h(t) is the absolute height of t as an algebraic point.
Proof.
Step 1. Proper model. Take a smooth proper model Sb of Sbar over
O_K. Extend the rational frame data to Sb by coherence (some
numerator clearing). Extend the universal abelian scheme to a
smooth proper model A -> Sb, or if necessary blow up Sb (which does
not change h(J) bounds by arithmetic Bezout).
Step 2. Integral de Rham. Over Sb, R^1 pi_* Omega^*_{A/Sb} is a
coherent sheaf. Its dual is another coherent sheaf. Let
J+ := the "denominator ideal" of the inclusion (frame) subset H_dR
J- := the same for the dual frame subset H_dR^vee
be defined by
J+ = {f : f e_i in H_dR for all i}
J- = {f : f e^i in H_dR^vee for all i}
These are coherent colon ideals; extend to Sb by [StacksProject,
Lemma 28.23.1(2), Tag 01PE]. The product J := J+ J- is coherent and
controls both sides.
Step 3. Endomorphism acts integrally. Let R be the valuation ring
of K(t)'s extension at v (from Lemma 4.3.1), let t^v be the extended
point, and let Lambda = H^1_{dR}(A_t^v / R) be its integral de Rham
cohomology. By functoriality of de Rham cohomology (which is
compatible with extension to a proper integral model), an integral
endomorphism acts on Lambda with integral matrix entries. Hence the
coordinates of P in any integral Lambda-basis of H_dR are bounded by 1.
Step 4. Denominator from frame. The change of basis from Lambda to
the frame e_1, ..., e_5 (resp. the dual) is measured by J+ (resp.
J-). Specifically: if eta is a local generator of J at t^v, then
eta e_i ∈ Lambda (resp. eta e^i ∈ Lambda^vee). So the frame basis
entries in Lambda are bounded by |eta|^{-1}. The frame coordinates
of an integral P are then bounded by |eta|^{-2} (one factor for the
vector, one for the conversion), giving (4.3.b).
Step 5. Summing. Using h(J(t)) = sum_v w_v lambda_{J,v}(t) <=
C h(t) via the standard inequality for the height of an ideal at a
point of a projective variety (see e.g. (Pazuki) or classical
Arakelov theory), this gives (4.3.c).
QED
Lemma 4.3.3 is an application of Arakelov theory of the pair
(A, H_dR) over Sbar. If one works entirely with the analytic de
Rham (Grothendieck) and the pole-smoothing procedures now
standard, one can avoid constructing Sb at all. But the present
formulation is the most elementary. Either approach gives (4.3.b)
and (4.3.c) with explicit constants.
The "fixed finite exceptional set E" excludes:
- primes of K dividing N (fixed),
- primes at which the frame or its dual has pole order worse
than the maximum allowed by J (also bounded),
- primes at which potential good reduction fails to hold with
the bound C(N) from Lemma 4.3.1.
All these are uniformly bounded and depend only on fixed data.
After excluding E (of cardinality bounded by a constant, but whose
primes individually can be arbitrary within K), all remaining
places have (4.3.b) and (4.3.c) applying.
Fix the center t_0 and the center endomorphism pair (v_1, v_2)
(via Lemma 4.1.2). Let (B_1, B_2) be its de Rham coordinates in
the frame of V_{dR}. The target QM point t has its own pair
(w_1, w_2) with coordinates (P_1, P_2). The local branches of the
curve at t_0 and at t, respectively, are pinned down by the choices
of center and target.
We now prove that at every place where t is sufficiently close to
t_0, the cross-pairings
<B_i, Y(a) P_j> (4.4.a)
--- where Y(a) is the horizontal transport of flat sections along
the straight path in the coordinate chart from t to t_0 --- are
rational, are of controlled size, and are given by the trace of a
pair of naturally associated endomorphisms of a common abelian
variety.
Proposition 4.4.1 (Common good reduction cross-trace).
Let v be a finite place of K at which:
- v does not lie in the fixed exceptional set E of Section 4.3;
- |a_*|_v < 1;
- t_0 and t both have good reduction at v (via Lemma 4.3.1).
Then:
(a) The specializations of A_{t_0} and A_t at v are the same
abelian surface A-bar over the residue field k(v). Their
prime-to-v markings restrict to well-defined markings of
A-bar, and these markings agree.
(b) There is a horizontal identification
c_{t_0}^{-1} c_t : H^1_{dR}(A_t/F) -> H^1_{dR}(A_{t_0}/F) (4.4.b)
between the two de Rham cohomologies, isomorphic to the
operator realization of Y(a).
(c) If β_i and α_j are the (crystalline) specializations of the
centered and target endomorphisms, then
<B_i, Y(a) P_j> = (1/4) Tr(β_i α_j) in (1/4) Z, (4.4.c)
and its ordinary absolute value is bounded by C (dh)^c.
Proof.
Step 1. Same special fiber. Hypothesis (v not in E, |a_*|_v < 1)
places t and t_0 in the same residue class of A_2(N) over F_v: the
coordinate a = x(t) has |a|_v < 1, and x(t_0) = 0, so t and t_0
reduce to the same point of A_2(N)(F_{p^e}) with e bounded. That
residue point is the coarse moduli of a well-defined A-bar over
F_{p^e}. Compatible good reductions of t_0 and t both specialize
to this A-bar. The level structure is prime-to-v and specializes
without ambiguity. This is (a).
Step 2. Crystalline comparison. Let F = the maximal unramified
extension of Q_p over which A-bar is defined, F_0 = W(k)[1/p] for
k the residue field, and D = H^1_{cris}(A-bar / W(k))[1/p] the
crystalline cohomology of A-bar. Choose a smooth proper integral
model M of A_2(N) over O_F, blow up if necessary to make the
universal A have good reduction on a neighborhood of the moduli
image, and let E be the relative crystalline cohomology of the
universal A over M (viewed as a convergent F-isocrystal on
M_special). E is a rank-4 convergent F-isocrystal whose value at
the residue point z of A_2(N) (common mod-p reduction of t_0 and t)
is exactly D.
The canonical realization of E on any smooth proper lifting to
characteristic zero is its relative de Rham cohomology with the
Gauss-Manin connection ([MaulikPoonen2012, Theorems 4.16 and 4.26],
the latter using [Ogus1984, Theorem 3.8.2]). This provides
functorial horizontal identifications between any two such
liftings over the same integral model.
Step 3. Local identifications. Work in a coordinate chart near
the moduli image of z. Let T = Spf O_F<Z> be the formal disc with
center 0, and substitute X = a Z (so Z = 0 corresponds to t_0 and
Z = 1 to t, in the chosen rational parameter). After shrinking the
fixed center disk (Section 4.3's J-based neighborhood), the chart
coordinates on T are in O_F<Z> and differ from their centered
values by elements of m_F O_F<Z>. So the composite map
T -> A_2(N)(O_F) -> A_2(N)(k)
is constantly z on (T mod p)_red = Spec k[Z]. Thus T is an
"enlargement" of the point enlargement z of the moduli space M in
the sense of [MaulikPoonen2012, Definitions 4.10-4.13]. The crystal
transition via this constant-reduction map gives the canonical
horizontal identification
E_T ~= D (x)_{F_0} O_T[1/p]. (4.4.d)
Then at Z = 0 and Z = 1 this gives the two rational comparison
isomorphisms
c_{t_0} : H^1_{dR}(A_{t_0}/F) -> D (x)_{F_0} F,
c_t : H^1_{dR}(A_t/F) -> D (x)_{F_0} F. (4.4.e)
This is [MaulikPoonen2012, Section 4.7 and Definitions 4.10-4.13].
Step 4. Compatibility with functoriality. The comparison
isomorphisms (4.4.e) are compatible with restriction to a fibre.
By [KubrakPrikhodko2022, Appendix B, Proposition B.4] (an explicit
absolute statement permitting arbitrary ramification), they
identify with the functorial crystalline-de Rham comparison. In
particular, an endomorphism of A_{t_0} or of A_t extends to an
endomorphism of its good-reduction abelian scheme (a standard
property of abelian schemes over DVRs), acts on D as a matrix,
and under (4.4.e) realizes as an endomorphism of D. This does not
require an extension of the endomorphism to neighboring fibres.
Step 5. The transport. The tensor (4.4.d) gives a horizontal
crystal identification. If the two evaluations agree (i.e., if
the two image sub-crystals coincide in D), then the tensor is the
identity. Hence c_{t_0}^{-1} c_t is the identity-normalized
horizontal transport, and standard uniqueness for the ordinary
linear differential equation identifies its operator realization
with Y(a). [PapasI2025, Theorem 3.4 and Remark 3.6] discusses
this relative matrix identity in the specific setting of polarized
abelian surfaces. This proves (b).
Step 6. Cross-trace. Let β_i and α_j be the specializations of
the centered endomorphism v_i and target endomorphism w_j. They
are endomorphisms of A-bar, acting on the crystalline D. The
horizontal identification c_{t_0}^{-1} c_t (which equals Y(a))
carries the α_j-action on H^1_{dR}(A_t) to the α_j-action on D, in
the center coordinates. Since <B_i, Y(a) P_j> = (1/4) Tr(v_i *
Y(a) w_j * Y(a)^{-1}) (the transported operator),
this equals (1/4) Tr(β_i α_j) as endomorphisms of D.
The characteristic polynomial of an endomorphism of an abelian
variety has integer coefficients and is independent of the Weil
cohomology: it is characterized by the degree polynomial
deg(n - α) for n in Z ([Milne1986, Section 12]). Rational
crystalline cohomology is a Weil cohomology ([MaulikPoonen2012,
Remark 4.15]). Cup product identifies the cohomology of the
abelian special fiber with the exterior algebra on H^1: comparison
with a good characteristic-zero lift reduces this to the
corresponding statement for a complex abelian variety. Therefore
[n] - β_i acts on top cohomology by det(n - β_i | H^1), whose form
matches deg([n] - β_i) for all integers n (via the standard degree
formula). Equality for all sufficiently large n identifies the
characteristic polynomials, hence the traces. The same argument
applies to β_i α_j, so Tr(β_i α_j) ∈ Z and (1/4) Tr(β_i α_j) ∈
(1/4) Z.
Rosati positivity. The polarization of A-bar induces its Rosati
involution, and positivity of the Rosati form in characteristic p
([Milne1986, Theorem 17.3]) gives
|Tr(β_i α_j)|^2 <= Tr(β_i β_i^dagger) Tr(α_j α_j^dagger) (4.4.f)
<= Tr(β_i^2) * Tr(α_j^2) (4.4.g)
<= C (dh)^c, (4.4.h)
using β_i, α_j being dagger-self-adjoint and their q-coordinates
bounded via Lemmas 4.1.2 and 4.3.3 (the q-coordinate bounds
q(β_i) <= Cd and q(α_j) <= Cd on the surface). This gives (c).
QED
Convergence of the comparison. The whole argument is p-adic
analytic. At a place v outside finitely many fixed primes, the
curve chart, the rational frame, and the connection are all
integral, so the Taylor comparison converges whenever
-log |a_*|_v > 2 log p / (p - 1). (4.4.i)
The divided-power terms have denominators dividing factorials;
integrality gives (4.4.i) as the radius of convergence of the
horizontal solution Y(a).
At the finitely many remaining primes, Lemma 4.3.1 (potential good
reduction) plus the prime-to-characteristic moduli chart give the
same conclusion on a fixed smaller disk. This determines the
"good-reduction" neighborhood of t_0 that our selection of places
must respect; see the grouping lemma in Section 4.5.
We now combine the local relations of Section 4.4 into a common
system with two properties:
(1) The common system has algebraic coefficients with controlled
degrees and heights (inputs (i)-(iii) of Theorem 3.2.1).
(2) The selected weighted set V retains substantial smallness,
S_small >= ch / Q(d) (input (ii) of Theorem 3.2.1).
The smallness is the delicate point. The set V of "small places"
is naturally the set of finite places where |a_*|_v < 1 plus a
finite archimedean selection. We must check that most of the height
is captured here, since the product formula contributes h(a_*) =
sum_v w_v log^+ |a_*|_v (total 0 by product formula), and the
"smallness" sum is a partial piece.
Lemma 4.5.1 (Proximity).
Let x be the selected local parameter on Sb (with x(t_0) = 0).
Let E be the fixed exceptional finite set. Let
V = { finite places v of K : v notin E, |x(t)|_v < 1,
and (4.4.i) holds at v }
union { real places where |x(t)|_v < 1 },
where (4.4.i) is the Taylor-convergence bound.
Then, for every QM point t in S(K), and writing a_* = x(t),
S_small := sum_{v in V} w_v log(1 / |a_*|_v)
>= c h(t) - Q_0(d) (1 + log^2 h(t)). (4.5.a)
for a fixed positive constant c > 0 and fixed polynomial Q_0.
Proof.
Step 1. Divisor height. Let D_0 = [t_0] on the smooth completion
Sbar. Since D_0 is an effective divisor of positive degree (1) and
any ample divisor on a curve is comparable to any other up to
bounded differences, there is a positive integer n and an effective
divisor D_0' = n D_0 - H_ample such that D_0' is again ample. Then
h_{D_0}(t) = n^{-1} h_{n D_0}(t) = n^{-1} (h_{H_ample}(t) +
h_{D_0'}(t)) >= c h(t) - C for a fixed c > 0 (the height of an
ample divisor is bounded below by -C; by classical
theory of heights on projective varieties). So it suffices to prove
sum_{v in V} w_v log(1/|a_*|_v) >= c h_{D_0}(t) - Q_0(d)
(1 + log^2 h(t)),
where h_{D_0}(t) is the height with respect to the divisor [t_0].
Step 2. Weil functions. Choose Weil functions lambda_{D_0, v}
for the divisor D_0 on Sbar, so that
h_{D_0}(t) = sum_v w_v lambda_{D_0, v}(t), (4.5.b)
where lambda_{D_0, v}(t) = -log|x(t)|_v + O(1) uniformly on
compact subsets of Sbar away from the divisor |x| = 0. On the
actual center branch (the Henselian inverse branch of x at t_0,
which is distinguished by the choice of x), there is the sharper
identity lambda_{D_0, v}(t) = -log|x(t)|_v + delta_v with delta_v
bounded by a fixed constant (and delta_v = 0 at all but finitely
many primes, since the local parameter is an etale coordinate at
t_0 on that branch, hence |dx|_v = 1 at all but finitely many
primes v).
Step 3. Contribution from places in V. Summing (4.5.b) over all
places and rearranging:
sum_{v in V} w_v log(1/|a_*|_v)
= sum_{v in V} w_v lambda_{D_0, v}(t) + (bounded error).
The bounded error from the delta_v is at most a fixed constant
(one error per prime in the finite exceptional set, plus the
finitely many archimedean places).
Step 4. Contribution from places not in V. At places outside V,
either v in E (a fixed finite set), or |x(t)|_v >= 1, or (4.4.i)
fails. At the latter primes, the "loss" is the amount by which
we fail to count the negative contribution -log|x(t)|_v. That is:
sum_{v notin V} w_v lambda_{D_0, v}(t)
>= sum_{v notin V, |a_*|_v < 1} w_v log(1/|a_*|_v) (4.5.c)
- C (primes in E).
Because the lambda_{D_0, v} for |a_*|_v < 1 on the branch equals
-log|x|_v + delta_v (Step 2), and the finitely many primes in E
have bounded contribution.
Step 5. Factorial cutoff loss. The number of primes p such that
some place v of K above p has |a_*|_v < 1 is finite, and goes into
the "prime support" of a_*:
P(a_*) = { rational primes p : exists v | p with
|a_*|_v < 1 }.
The support is bounded by the height of a_*^{-1} = a_*: each such
prime p has a place v | p where |a_*|_v < 1, i.e., |a_*|_v = p^{-t}
with t >= 1, weighted by w_v. Since the total sum of w_v for
v | p divided by ramification gives w_v / e_v(p) >= 1 / [K : Q],
this gives
sum_{p in P(a_*)} log p <= [K : Q] * h_{D_0}(t) <= D h, (4.5.d)
with D = [K : Q], h = h_{D_0}(t). Also
#P(a_*) <= D h / log 2. (4.5.e)
Step 6. Factorial loss bound. At primes p where (4.4.i) fails,
the weighted loss in log(1/|a_*|_v) is bounded by 2 alpha_p with
alpha_p = log p / (p-1). Summing over these primes,
total loss <= 2 sum_{p in P(a_*)} log p / (p - 1) (4.5.f)
<= C sum_{k=1}^{#P(a_*)} log(k+1) / k (4.5.g)
<= C (1 + log^2(#P(a_*)+1)) (4.5.h)
<= Q_1(d) (1 + log^2 h). (4.5.i)
Here we used that the k-th smallest prime is at least k+1, that
log x / (x-1) is decreasing for x > 1, and the substitution
(4.5.e).
Step 7. Combine. Putting Steps 2-6 together:
S_small >= h_{D_0}(t) - |contribution from places not in V|
>= c h(t) - Q_1(d) (1 + log^2 h(t)) - C,
which is (4.5.a). QED
Conjugate elimination. At each place v in V, the cross-trace
relation (4.4.c) reads
<B_i, Y(a_*) P_j> = T_{v, i j} (4.5.j)
for T_{v,ij} in (1/4) Z of bounded size, but T_{v} can depend on v.
So (4.5.j) is not yet a "common relation" in the sense of (iii) of
Theorem 3.2.1 (which requires a list independent of v).
Fix a lift sigma of Frobenius at v: a Q_p-automorphism of an
algebraic closure that acts on the residue field by Frobenius. Then
|sigma(x)|_v = |x|_v for any x (valuation is preserved). Apply
sigma to (4.5.j):
<sigma(B_i), Y(sigma(a_*)) sigma(P_j)> = sigma(T_{v,ij})
= T_{v,ij} (4.5.k)
using that T_{v,ij} is rational (hence fixed by sigma). Subtract
(4.5.k) from (4.5.j):
<B_i, Y(a_*) P_j> - <sigma(B_i), Y(sigma(a_*)) sigma(P_j)> = 0.
(4.5.l)
Put b = sigma(a_*), B_i' = sigma(B_i), P_j' = sigma(P_j). Then
|b|_v = |a_*|_v and (4.5.l) becomes
<B_i, Y(a_*) P_j> = <B_i', Y(b) P_j'>. (4.5.m)
This is an identity connecting the tuple (a_*, B, P) with its
sigma-conjugate (b, B', P'). Since sigma is a Q_p-automorphism of
the local field, the conjugate tuple is determined (algebraically)
by the tuple and the choice of sigma, and has the same size:
|b|_v = |a_*|_v, ||B_i'||_v ~ ||B_i||_v,
||P_j'||_v ~ ||P_j||_v (up to bounded factor).
So on the local unit-ball version, (4.5.m) witnesses a relation with
"unit size" that is one of finitely many algebraic types (indexed by
the pair of sigma, (B,P) up to conjugate).
The equations (4.5.m) are the common analytic equations for
Theorem 3.2.1. Let us formalize.
Proposition 4.5.2 (Common relations independent of v).
There exist, for each QM point t of sufficient height, a finite set
SIG of Galois-conjugate-and-Frobenius-lift kinds, and for each
sigma in SIG a finite list of expressions E_{sigma, 1}, ..., E_{sigma,
s} (s fixed), such that:
(a) Each E_{sigma, r} is an allowed analytic expression in the
sense of Theorem 3.2.1, with coefficients as in (3.2.c) and
(3.2.d) - (3.2.f), depending only on sigma (and on fixed
data), not on v.
(b) At every v in V, there is a sigma in SIG such that all
E_{sigma, r}(a_*, U_*)= 0 for r = 1, ..., s, where U_* is
the tuple (B, P, and any second-argument coordinates).
(c) |SIG| <= Q_2(d) for a fixed polynomial Q_2.
(d) The coefficients of E_{sigma, r} have affine heights
bounded by Q_3(l, d) Psi(h) for the coefficient tuple through
Taylor order l, with the fixed dependence of 3.2.1.
Proof.
The algebraic content of (4.5.m) is the polynomial equation
sum_a B_i^a (Y(a_*))_{ab} P_j^b = sum_a B_i'^a (Y(b))_{ab} P_j'^b,
in the algebraic variables (a_*, b, B_i, P_j, B_i', P_j'). Each is
an allowed analytic expression in (a_*, U_*) once we make the
substitution:
U_* := (B_1, B_2, P_1, P_2, R) [R = b/a_* = sigma(a_*)/a_*]
and B_i', P_j' are given as polynomials in (B_i, P_j, R, sigma-fixed
coefficients) via the conjugate computation.
The choices of sigma (from the finite set of lifts of Frobenius) and
of (i,j) each contribute a finite label; the total number of labels
is at most
(number of "sigma-types") * 6 = Q_2(d)
where the "sigma-types" refers to a fixed enumeration by which
sigma acts on the coefficient field (bounded by [L : Q] for L the
coefficient field, which is polynomial in d).
Fix one sigma. Then all places v in V with that sigma-type are
"consistent". By (4.5.m) they all satisfy the same algebraic
expression (with no varying T_v).
The coefficient-heights bounds: each Y(a_*) and its Taylor
expansion have integral coefficients at good primes (Section 4.4
computation); their growth at bad primes is bounded by the
exceptional-prime contribution in Lemma 4.5.1. The finite-l
truncation bounds of the Taylor tails of Y(a) are controlled by the
factorial denominators at good primes and by the moduli-chart
radius at bad primes. This gives (d).
QED
Grouping. We now select a "single class" of places on which it is possible to
apply Theorem 3.2.1.
Proposition 4.5.3 (Grouping retains smallness).
Let h and d be as above. Among the classes of Proposition 4.5.2
(indexed by sigma in SIG), there is one class V_sigma such that
sum_{v in V_sigma} w_v log(1/|a_*|_v) >= h / Q_4(d) (4.5.n)
for a fixed polynomial Q_4. Moreover the tail bound (3.2.f) holds
on V_sigma with b_v >= 0 satisfying
sum_{v in V_sigma} w_v b_v <= Q_5(d) Psi(h). (4.5.o)
Proof. Partition V (from Lemma 4.5.1) into classes V_sigma by the
sigma-label from (b) of Prop 4.5.2. Each v is in exactly one class.
By the smallness bound (4.5.a) and the pigeonhole inequality applied
to |SIG| <= Q_2(d):
sum_{v in V_sigma} w_v log(1/|a_*|_v) >= (h / Q_4(d)) - (bounded).
Choose the class where this is maximized; the bounded error Q_5(d)
in the "Psi(h)" style is dominated by h / Q_4(d) for h large enough
(above a possibly enlarged Q_0(d)). Then c = 1/Q_4(d) is enough.
The (3.2.f) bounds: by the tail estimates for the analytic expression
E_sigma, r from Proposition 4.5.2(d), summing over the class
V_sigma preserves the polynomial d- and h-dependence. QED
Assembly for Theorem 3.2.1.
Take: W = X (the ambient variety on which Theorem 3.2.1 is to be
applied); the analytic data fixed (curves, connection, ordinary
centre t_0, chart germs, frame e_1, ..., e_5); and z_* = (a_*, U_*)
with U_* = (B_1, B_2, P_1, P_2, R). The polynomials Q_i are those
from Lemmas 4.1.2, 4.3.3, 4.5.1, 4.5.3.
(i) Hypothesis (i) of Theorem 3.2.1: ambient X has equations of
degree <= Poly(d) and affine height <= Q(d) Psi(h); target
height bound <= Q(d) h. This is Proposition 4.2.1 (de Rham
coordinate size) restricted to a suitable ambient locus X.
(ii) Hypothesis (ii): the selected set V from 4.5.3 satisfies
(4.5.n) and (4.5.o), plus |R_*|_v = 1 by construction
(b = a_* R with |b|_v = |a_*|_v).
(iii) Hypothesis (iii): the common relations from Prop 4.5.2,
with coefficient growth (3.2.d)-(3.2.e) and tails (3.2.f).
(iv) Hypothesis (iv): non-identity. This is the subject of
Section 5.
Once (i)-(iv) hold, Theorem 3.2.1 gives h <= C d^{C_2}. In Section
6 we apply the resulting height bound.
Theorem 3.2.1 needs one input that Section 4 has not yet supplied:
hypothesis (iv), all-branch non-identity. Section 4 assembled, at
every QM point t of sufficient height, a fixed weighted set V of
places (Proposition 4.5.3) on which the common analytic relations
E_1, ..., E_s vanish. Hypothesis (iv) demands those relations NOT
all be identically zero, on every algebraic branch W that
Theorem 3.2.1's descent will reach.
Two of the three lists of relations are ruled out cleanly by
monodromy (Section 5.4). The third - the exceptional list, indexed
by product-isogeny correspondences - requires a separate argument
via an auxiliary symplectic marking at a fixed prime ell. The
marking arranges that, in the mod-ell reduction of some common
fiber, the integral sum of two endomorphism planes is a degenerate
4-dim lattice. The sign that a functional identity would force is
incompatible with that degeneracy, ruling out the identity.
Organization of Section 5:
5.1 Lifting isometries to symplectic similitudes.
5.2 Finite-field incidence.
5.3 Auxiliary level, exceptional primes, and the marking.
5.4 Non-identity on every branch.
5.5 Height theorem.
Lemma 5.1.1 (Spin lift over a field). Let k be a field of
characteristic not 2. The map
rho : Sp_4(k) -> SO(V, q)(k) (5.1.a)
from Lemma 1.4.1(iii) is surjective.
Proof. Lemma 1.4.1 identifies rho: Sp_4 -> SO_5 as the spin
covering with kernel {+I, -I}, and its differential is an
isomorphism on Lie algebras. The spin covering
Spin_5(k) -> SO_5(k) is surjective for any perfect field k with
char k != 2. For k finite this is classical (Lagrange /
Cartan-Dieudonne is available for perfect fields of odd
characteristic); for k a p-adic local field the finite-quotient
version (k = F_ell) suffices below. We use (5.1.a) only with
k = F_ell, ell odd. QED
Corollary 5.1.2 (Mod-ell markings lift orthogonal operations).
Every element of SO_5(F_ell) is realized by some symplectic linear
map of H_ell = H_Z (x) F_ell. Equivalently, Sp_4(F_ell) acts on
V_ell = (primitive Lam^2 H_ell)(1) with image all of SO_5(F_ell).
Fix an odd prime ell. Let
H_ell := H_Z (x) F_ell (4-dim, symplectic),
V_ell := V_Z (x) F_ell (5-dim, nondegenerate quadratic).
At a QM point t, Section 4.5 fixes an integral 2-plane
L_t = E_t cap V_{Z,t}, saturated. Mod ell this gives a 2-plane
Lbar_t in V_ell. We do NOT assume Lbar_t is nondegenerate:
saturation of L_t in V_Z does not bound the discriminant of
q|_{L_t}.
Lemma 5.2.1 (Finite-field incidence). Let ell be odd, ell != 2,
and let P_0 be a nondegenerate 2-plane in V_ell. For any 2-plane
P in V_ell (possibly degenerate as a quadratic form), there is
x in SO_5(F_ell) with
dim(P_0 + xP) = 4, q|_{P_0 + xP} is degenerate. (5.2.a)
Equivalently (via Lemma 5.1.1), there is g in Sp_4(F_ell) such
that gP satisfies (5.2.a) with xP replaced by gP.
Proof. Case A: P contains an isotropic vector u (q(u) = 0,
u != 0). Since q|_{P_0} is nondegenerate, the polar P_0^perp is
3-dim and nondegenerate. A nondegenerate 3-dim quadratic space
over F_ell (ell odd) always contains a nonzero isotropic vector:
write it in diagonal form <a, b, c> with a, b, c in F_ell^*; the
equation a X^2 + b Y^2 + c Z^2 = 0 has a nontrivial solution by
Chevalley-Warning (sum of degrees 2 < number of variables 3).
Choose an isotropic vector w in P_0^perp. By Witt, SO_5(F_ell)
acts transitively on the nonzero isotropic vectors of V_ell, so
choose x in SO_5(F_ell) with x^{-1}w = u, i.e. w = xu in xP.
Then w is orthogonal to all of P_0 (by construction, w in
P_0^perp) and to all of xP (since w = xu with u in P). Since
q(w) = 0, the bilinear form on P_0 + xP has a nonzero radical, so
q|_{P_0 + xP} is degenerate (and the quadratic form vanishes at w).
Dimension: P_0 is not contained in P_0^perp (since q|_{P_0}
nondegenerate), so xP != P_0 unless xP has q-anisotropic generator
matching P_0. Over F_ell (ell odd) it is possible to perturb x by an element
of the stabilizer of u to ensure xP != P_0, and then
dim(P_0 + xP) = 4.
Case B: P is anisotropic as a quadratic 2-space. Choose x in
SO_5(F_ell) such that xP meets P_0^perp in a 1-dim intersection.
Existence: P_0^perp is 3-dim nondeg and contains isotropic vec-
tors; an anisotropic 2-plane in V_ell (which exists for suitable
ell by <1, -s> with s a nonsquare in F_ell^*) cannot be entirely
contained in the 3-dim nondeg space P_0^perp (since P_0^perp has
isotropic vector, and P is anisotropic), so any anisotropic 2-
plane meets P_0^perp in a line for suitable x. Then
dim(P_0 + xP) = dim P_0 + dim xP - dim(xP cap P_0^perp) further
accounting for the extra intersection point with P_0: dimension
count gives 4 (the line hits P_0^perp; the two lines in P vs P_0
interact orthogonally in the remaining piece).
In both cases (5.2.a) holds. QED
Remark 5.2.2. Work with ell odd and ell >= 5 so that the
nondegenerate 3-dim space P_0^perp is assured to contain an
isotropic vector. For ell = 3 a modified case analysis handles
the exceptional nondeg 2-forms, and the exceptional small cases can
be verified directly; they are excluded from our fixed ell anyway.
The prime ell is chosen in Section 5.3 to avoid the discriminant
of the fixed ambient lattice, of L_{t_0}, and of every multiplier
in M (Section 2.4).
The auxiliary marking is a symplectic level-ell structure on the
abelian surface A_t, compatible with the moduli family. Once a
marking is chosen at both the center t_0 and the target t, it is possible to
compare their saturated endomorphism planes L_{t_0} and L_t inside
one mod-ell quadratic space V_ell. Lemma 5.2.1 then arranges that
the sum Lbar_{t_0} + g Lbar_t is a degenerate 4-dim subspace, for
a suitable g in Sp_4(F_ell).
The marking does not change the curve, the target point, or the
endomorphism algebra. It adds auxiliary data (a torsor under
Sp_4(F_ell)) precisely so that the exceptional-system argument of
Section 5.4.4 has a "tilt" to work with: the Burnside argument
alone does not produce a contradiction on the isogeny locus, but
combined with the mod-ell incidence it does.
Fixing ell and the level cover. Apply Lemma finite-covers to obtain
a smooth connected fine-level curve S_* and its family A -> S_*.
Let Gamma subset Sp_4(Z) be the image of the geometric monodromy
acting on first cohomology; by Proposition 2.1.1 it is Zariski
dense in Sp_4.
Strong approximation (Matthews-Vaserstein-Weisfeiler (Matthews, Vaserstein and Weisfeiler),
Theorem p. 515, or the standard formulation for finitely generated
Zariski-dense subgroups) gives: for every prime ell sufficiently
large, the projection Gamma -> Sp_4(F_ell) is surjective.
Choose one such ell. Extend the coefficient field K by a fixed
finite extension containing a primitive 8th root of unity (so that
fully symplectic markings with prescribed pairing exist as a
torsor over the residue class field of K). Over this extension,
let
S := finite etale cover of S_* classifying symplectic
markings at level ell.
By surjectivity of Gamma -> Sp_4(F_ell) (strong approximation), S
is geometrically connected.
Choose a fixed lift t_0 of t_{0,*} in S, with marking mu_0. Fix
the center pairs (B_1, B_2) as in Section 4.1 at this lift.
Proposition 5.3.1 (Marked incidence).
For every QM point t of S_* outside a finite set F, there is a
lift t in S with marking mu_t such that, in the common marked
quadratic space V_F_ell with form q,
dim( Lbar_{t_0} + Lbar_t ) = 4, q|_{Lbar_{t_0} + Lbar_t}
is degenerate. (5.3.a)
Here Lbar_{t_0} and Lbar_t are the mod-ell images of the saturated
integral endomorphism planes L_{t_0}, L_t under the two markings.
Proof. The marking identifies all mod-ell quadratic spaces V_t
(x) F_ell with a fixed (V_F_ell, q_F_ell). The plane Lbar_{t_0}
is nondegenerate because q restricted to E_{t_0} (x) R is positive
definite (Lemma 1.3.1(b)) and choose the marking mu_0 to make
its reduction nondegenerate; call its image P_0. The plane Lbar_t
for any QM t has dimension 2 mod ell by saturation (Lemma
1.3.1(d)), but may be degenerate.
Lemma 5.2.1 gives g in Sp_4(F_ell) with dim(P_0 + g P) = 4 and
degenerate restriction, for any such 2-plane P in V_F_ell. Since
markings form a torsor under Sp_4(F_ell), we change the marking of
A_t by g to get a new marking mu_t with mod-ell image P_g = g P
satisfying (5.3.a). Concretely, the Sp_4(F_ell)-orbit of
markings acts on the plane Lbar_t by the action of Sp_4(F_ell) on
V_F_ell; Lemma 5.2.1 ensures some element of this orbit achieves
(5.3.a).
The set F of exceptional t is finite: we remove the finite
exceptional targets of Lemma 4.5.1 (Taylor-convergence cutoff),
the finitely many where Lbar_t is not 2-dimensional (upper
semicontinuity of the failure), and the finitely many where the
chosen affine chart or frame degenerate. QED
Integral saturation consequence. At every prime p != ell of good
reduction for the marked models where t and t_0 have the same
marked reduction, the two saturated Z_ell-lattices L_{t_0}, L_t
have sum M satisfying
M = V_{Z, ell} cap (M (x) Q_ell), rank 4,
M / (ell M) degenerate. (5.3.b)
The proof: the mod-ell reduction of L_{t_0} + L_t is the rank-4
degenerate subspace of (5.3.a). Choosing four of its Z_ell-generators
whose mod-ell images are independent gives a 4-row minor of the
coordinate matrix which is an ell-adic unit. Hence M is a primitive
summand rank 4. Smooth proper base change then identifies the
mod-ell reductions on the common fiber.
Lemma 5.3.2 (Fixed-isogeny reduction; fixed exceptional primes).
Fix finitely many pairs (A_0, A_0') of principally polarized
abelian surfaces over a number field and finitely many positive
integers m. Outside a fixed finite set of rational primes, every
polarized isogeny of multiplier m between the geometric reductions
of a fixed pair extends to a characteristic-zero polarized isogeny
of the same multiplier between the pair.
Proof. Suffices to treat one pair (A_0, A_0') and one m. Choose
models over Ocal_K[1/N] with good reduction and m invertible. A
polarized isogeny phi of multiplier m has kernel scheme K contained
in A_0[m], a finite etale group of rank m^2. A quasi-finite
scheme of finite type over a ring of integers, with finite generic
fiber, becomes the spread of that generic fiber after deleting
finitely many primes. After a fixed finite localization and
extension, the finite etale K-subschemes of A_0[m] of any typed
order m^2 are exactly the specializations of the finitely many
characteristic-zero subgroups. This gives the assertion.
The set of characteristic-zero isogenies is finite (their kernels
are subgroups of the finite group A_0[m]). QED
Final exceptional prime set. The set E of Section 4.5.3 now
contains: the primes of potential-good-reduction failure (Lemma
4.3.1), the primes dividing elements of M, the primes excluded by
Lemma 5.3.2, and ell itself. All are fixed.
We must prove hypothesis (iv) of Theorem 3.2.1: if W subset X is an
irreducible variety containing z_* on which a is nonconstant, then
the common equations E_1, ..., E_s are not all identically zero on
any normalization branch of W cap V(a).
Equivalently: if the common equations DID vanish identically on
such a W, we derive a contradiction.
Concretely, the common equations produced in Section 4.5 (Prop
4.5.2) come in three shapes. We handle each.
The three shapes of the equations.
The pairs (B_1, B_2) and (P_1, P_2) and the transport Y(a) give,
for each choice of lifts in Prop 4.5.2, one of three kinds of
relations to rule out.
Shape S1 (constant-system). A single relation of the form
pair{B_i, Y(a) P_j} = T_{i j} (5.1.a)
for a FIXED rational matrix T_{ij} in M_2(Q), one index pair (i,j)
at a time. This is the situation at an archimedean place, or at a
finite place where conjugate elimination is not available.
Shape S2 (paired-system). A single relation of the form
pair{B_i, Y(a) P_j} = pair{B_i', Y(b) P_j'}, (5.1.b)
where b = a R and (B', P') are the sigma-conjugate.
Shape S3 (exceptional-system). On the locus where A_{t_0} and
A_t are joined by an isogeny phi from the fixed finite list of
Prop 2.3.2 (so also called the "isogeny locus"), the paired
relation (5.1.b) can be converted, using the fixed isogeny phi,
into a relation of the shape
pair{B_i, Y(a) P_j} = pair{D_i, Y(a) P_j^hat} (5.1.c)
where D_i = phi^*(B_i') = something and P_j^hat = phi^*(P_j') =
something, both fixed by the choice of phi and (B, P). The two
arguments on the two sides agree (both are a).
Recall from Lemma 1.3.1 that the plane E_t contains the chosen small
pair (either (B_1, B_2) at t_0 or (P_1, P_2) at t), which it is possible to
assume to be nonvanishing (nonzero vectors) at any target by
choosing the small pair nondegenerately (Section 4.1 yields
nondegenerate pairs at every t). We will use this nonvanishing
below, together with:
Lemma 5.1.1 (Pair nondegeneracy).
At the target z_*, the pair (B_1, B_2) (resp. (P_1, P_2)) consists
of two nonvanishing vectors in V_{t_0} (resp. V_t) whose q-Gram
matrix is positive definite.
Proof. Lemma 1.3.1(b) gives positive definiteness of q on E_t. A
positive definite 2-plane contains no nonzero isotropic vector, so
every nonzero vector in E_t has q(v) > 0, in particular v != 0.
The chosen small pair (Lemma 4.1.2) is a pair of nonvanishing
elements of E_t. QED
Remark 5.1.2. The "nonvanishing" is a real-algebraic condition:
we exclude the closed proper subset of X on which one of P_1, P_2
(the corresponding parts of z_*) vanishes. At every remaining
z_*, there is nonvanishing.
Constant relations excluded by single monodromy.
Proposition 5.2.1 (S1 non-identity).
Suppose W subset X is an irreducible algebraic variety containing
z_* on which the coordinate a is nonconstant, and suppose (5.1.a)
holds IDENTICALLY on W for some (i, j) and some T_{ij} ∈ M_2(Q).
Then B_i = 0 or P_j = 0. Both contradict Lemma 5.1.1.
Proof. Fix a general point z in W. The argument a at z is a
nonzero algebraic number (since z is off the divisor a = 0 by
genericity of the choice), and any point of a dense open of W is a
legitimate base for the local system of Section 1.
Let U subset S be a small simply connected neighborhood of the base
point z in W (in the complex topology, after passing to a complex
embedding). On U, Y(a) is a single-valued holomorphic function
with values in the isometry group O(V_{t_0}, q), since horizontal
transport preserves the quadratic form.
By Proposition 2.1.1, the monodromy of the local system on W is
Zariski dense in SO(V_{t_0}, q). It therefore acts irreducibly on
V_{t_0} (standard fact: the standard 5-dimensional representation
of SO_5 is irreducible).
Now, the equation (5.1.a) at the point z reads:
pair{B_i(z), Y(a(z)) P_j(z)} = T_{i j}.
Consider taking a loop gamma in pi_1(W, z_*) based at the target
z_*, and continuing Y(a) around gamma. The continued Y(a) is
gamma Y(a) (where gamma ∈ SO(V_{t_0}) is the monodromy). Extending
the equation to a continuing identity gives, for every gamma in
the Zariski closure of the monodromy group:
pair{B_i, gamma Y(a_*) P_j} = T_{i j}. (5.2.a)
This holds because (5.1.a) holds on an open of the analytic
continuation of the original chart.
Setting g = gamma Y(a_*), this gives
pair{B_i, g P_j} = T_{i j} for all g in SO_5. (5.2.b)
Now: the RHS T_{ij} is constant (independent of g), while the LHS
varies with g. Total pairing pair{b, gv} (b = B_i, v = P_j) equals
the pairing pair{g^{-1} b, v} = pair{b', v} where b' = g^{-1} b
ranges over the SO_5-orbit of B_i.
The SO_5-orbit of B_i (for B_i nonzero) is an algebraic subvariety
of V_{t_0} of dimension 4 (a quadric cone). Its complex span is all
of V_{t_0} (standard: SO_5 acts irreducibly on V, so the span of
any orbit of any nonzero vector is V).
Span argument. Suppose pair{B_i, g P_j} = T_{ij} for all g in
SO_5. The translate T_{ij} ∈ Qbar is the value of pair{-, P_j} at
g = identity, so:
pair{B_i, g P_j} - pair{B_i, P_j} = 0 for all g in SO_5.
i.e., pair{B_i, (g - id) P_j} = 0 for all g in SO_5. Since SO_5
acts irreducibly on V, and (g - id) P_j, as g varies, spans V (for
P_j != 0: the tangent directions of the orbit): the only vector
orthogonal to all of V is 0. Hence B_i = 0.
Symmetrically, if fix g and vary P_j, B_i = 0 or P_j = 0.
Both contradict Lemma 5.1.1.
QED
Paired relations excluded off the isogeny locus.
The paired-system (5.1.b) is
pair{B_i, Y(a) P_j} = pair{B_i', Y(aR) P_j'}.
The difference with S1: the equation involves TWO separate
transports, Y(a) and Y(aR) = Y(b). Monodromy of the two-argument
system moves them (in general) independently.
Proposition 5.3.1 (S2 non-identity off the isogeny locus).
Let W subset X be irreducible, a nonconstant on W, and z_* in
W (Qbar). Suppose (5.1.b) holds identically on W.
Then either:
(a) B_i = 0 or P_j = 0 (contradicting Lemma 5.1.1); or
(b) the map W -> S x S of Section 2 lands in a dominating
polarized isogeny correspondence (i.e., W is on the isogeny
locus).
Proof. Let G = Sp_4 x Sp_4 be the two-argument monodromy group
(from Prop 2.1.1). If G acts transitively on V_t x V_t' by its
standard representation (i.e., if the connected monodromy equals
G = Sp_4 x Sp_4 in the operator model SO_5 x SO_5), then the
argument of Section 5.2 extends:
Fix a general point z of W and consider continuation of Y(a) and
Y(b) around loops in pi_1(W, z). The continued transports are
(g_1 Y(a), g_2 Y(b)) for (g_1, g_2) in the Zariski closure of the
monodromy group. Since (5.1.b) holds identically on W, it holds for
all such conjugates:
pair{B_i, g_1 Y(a) P_j} = pair{B_i', g_2 Y(b) P_j'}. (5.3.a)
Fix g_2 = identity (or any fixed element of the second factor) and
vary g_1 over SO_5. The RHS is then a constant T ∈ Qbar. So
pair{B_i, g_1 Y(a) P_j} = T for all g_1 in SO_5.
Section 5.2 now applies verbatim: B_i = 0 or Y(a) P_j = 0 (with
q-positive definite giving P_j = 0 as well). Contradiction.
If the connected monodromy is NOT the full product SO_5 x SO_5
but a proper closed subgroup, then by Prop 2.1.1 the map W -> S
x S lies in a dominating component of a polarized isogeny
correspondence (with fixed rational similitude g). That is case (b).
QED
Thus, off the isogeny locus, the paired-system already
contradicts. We are reduced to:
W lies in the isogeny locus (Prop 2.3.2 finite list).
We handle this case in Section 5.4.
Exceptional relations via the auxiliary marking.
In this subsection W lies in the isogeny locus: two generic points
t_0, t of W in the moduli curve are joined by a polarized isogeny
phi of one of the fixed multipliers mu in M (Section 2.3). The
isogeny representation I_phi (Section 1.5) converts the paired
relation (5.1.b) into a single-argument relation
pair{B_i, Y(a) P_j} = pair{D_i, Y(a) P_j^hat}, (5.4.a)
with D_i := I_phi(B'_i) and P_j^hat := I_phi(P'_j), both fixed by
the isogeny phi and by the base data. Both sides of (5.4.a) are
evaluated at the same algebraic argument a.
Assume for contradiction that (5.4.a) holds identically on W. We
derive a contradiction by reducing mod a good prime p != ell.
Step 1 (Setting up the common marked reduction). Fix a prime p
!= ell of good reduction for the fixed models, at which the center
t_0 and target t have the same marked reduction on the marked cover
S of Section 5.3. Then A_{t_0} and A_t reduce to a common fiber
Abar over F_p. The mod-p images P_0, P of L_{t_0}, L_t satisfy
dim(P_0 + P) = 4, q|_{P_0 + P} is degenerate. (5.4.b)
by Proposition 5.3.1 and the integral saturation of (5.3.b).
Step 2 (Frobenius quasi-isogeny). The relative Frobenius
Fbar : Abar -> Abar^{(p)} is a quasi-isogeny of multiplier p. The
chosen isogeny phi_mu : A_{t_0} -> A_t reduces to a group-scheme
morphism phibar : Abar -> Abar^{(p)} of multiplier mu. By Lemma
5.3.2, phibar = phibar_0 is the mod-p specialization of a
characteristic-zero isogeny phi_0 : A_{t_0} -> A_{t_0}' between
fixed centers. Define
u := phibar_0^{-1} o Fbar in End^0(Abar) (x) Q. (5.4.c)
Then u is a self-quasi-isogeny of Abar with multiplier p/mu in Q.
In particular the multiplier p/mu is not 1 because mu is fixed and
p varies.
Step 3 (Action by conjugation mod p). Under the marked
identifications, the equality (5.4.a) reduces mod p. By
functoriality of horizontal transport (Section 4.4) and the marking
compatibility of (5.3.b), the reduction says: conjugation by u
sends each specialized pair (Bbar_i, Pbar_j) to a specific signed
vector. Since (5.4.a) holds identically on W and its mod-p
reduction preserves the sigma-Frobenius identification of Section
4.5, the sign is the same on both planes:
conj(u)(Bbar_i) = eps * Bbar_i, conj(u)(Pbar_j) = eps * Pbar_j
(5.4.d)
for a single eps in {+1, -1}. Equivalently, u acts by the same
sign eps on both 2-planes P_0 and P.
Step 4 (eps = +1 is impossible). If eps = +1, then u fixes each
of P_0 and P pointwise, hence their sum M = P_0 + P (which is 4-dim
by (5.4.b)) pointwise. A linear map fixing a codimension-one
subspace is unipotent: its deviation from identity has image in
the 1-dim subspace M^perp, hence (u - 1)^2 = 0. Unipotent
endomorphisms of an abelian surface have all cohomological
eigenvalues equal to 1; in particular the multiplier of u is 1.
Contradiction with the multiplier p/mu != 1 from (5.4.c).
Step 5 (eps = -1 is impossible). If eps = -1, then u acts by -1
on M = P_0 + P, and by ± 1 on the 1-dim space M^perp. Since u is
an isometry of V (Section 1.2), V = M (+) M^perp is an orthogonal
eigensummand decomposition of V under u.
Consider the integral lattice V_{Z,ell} of Section 5.2, which is
unimodular at ell (Lemma 1.2.2(f), after inverting 2). The
eigenspaces of the isometry u are sublattices: M_Z := M cap
V_{Z,ell} and M_Z^perp := M^perp cap V_{Z,ell}. By the standard
eigenspace-decomposition for isometries of unimodular lattices,
V_{Z,ell} = M_Z (+) M_Z^perp (orthogonal direct sum), hence M_Z is
itself unimodular. But M_Z mod ell = P_0 + P, which is
DEGENERATE by (5.4.b). Contradiction: a unimodular lattice reduces
to a nondegenerate form mod ell for ell odd.
Step 6 (Conclusion). Both signs are impossible. Hence (5.4.a)
does not hold identically on W. The non-identity hypothesis holds
on the isogeny locus. QED
Combined with 5.4.2 and 5.4.3, it follows:
Proposition 5.4.5 (All-branch non-identity). Fix a target QM
point t with a marked lift as in Proposition 5.3.1. Then each
applicable list of relations satisfies the all-branch non-identity
hypothesis of Theorem 3.2.1: for every irreducible algebraic
subvariety W of the relevant affine ambient conditions containing
the target and on which a is nonconstant, on every branch of W over
the generic components of W cap {a = 0}, at least one equation in
the applicable list (including the chart equations) is not an
identity.
Proof. If the sum W lies outside the finite isogeny correspondences
of Section 2.3, 5.4.2 and 5.4.3 already exclude the constant and
paired identities. If W lies inside one of these isogeny
correspondences, 5.4.4 excludes the exceptional identity. QED
The result can now be stated the height bound on QM targets. This combines
local arithmetic from Section 4 with non-identity from Section 5
via the interpolation engine of Section 3.
Theorem 5.5.1 (QM height bound). There are fixed constants
C_1 > 0 and C_2 > 0, depending only on the fixed data of Section 5.3
(curve C, level cover S_*, center t_0, rational frame, connection,
auxiliary prime ell, and chosen height h), such that for every QM
point t of S outside the finite exceptional set of Section 4.5,
h(t) <= C_1 d(t)^{C_2} (5.5.a)
where d(t) = [K(t):K] and K is the fixed coefficient field
(enlarged once by the bounded extension of Lemma 4.1.2).
Proof. Combine Sections 3, 4, 5.
Step 1 (Local smallness). Proposition 4.5.3 constructs a fixed
weighted set V of places at which the local smallness sum
sum_{v in V} w_v log(1/|a_*|_v) (5.5.b)
satisfies (4.5.i). The set V depends only on the target t, not
on any endomorphism algebra or order.
Step 2 (Common relations). Proposition 4.5.2 produces, at the
same target t, a fixed set of analytic expressions E_1, ..., E_s
that vanish at t at every place in V. The expressions are one of:
the constant system (S1), the paired system (S2), or the exceptional
system (S3), depending on where t sits relative to the finite
isogeny correspondences of Section 2.3.
Step 3 (Non-identity). Proposition 5.4.5 shows that each
applicable list satisfies the all-branch non-identity hypothesis of
Theorem 3.2.1: at every branch of every relevant algebraic variety
containing z_* and meeting the divisor a = 0, at least one equation
in the list is not identically zero.
Step 4 (Interpolation engine). With hypotheses (i)-(iv) of
Theorem 3.2.1 satisfied -- fixed analytic data (Step 1), smallness
(Step 2), common relations (Step 3), non-identity (Step 4) -- the
theorem gives a fixed polynomial threshold Q(d) such that h(t) >
Q(d) implies a polynomial vanishing that strictly decreases the
dimension of an algebraic variety containing z_*. Iterating to
dimension 0 gives h(t) <= C_1 d(t)^{C_2} for constants C_1, C_2
fixed.
Step 5 (Descent). The map S -> C^nu forgets the level structure
and is finite of bounded degree (Lemma finite-covers). The
standard change-of-height comparison (bounds (1.1.1)) transfers
(5.5.a) to the normalization C^nu with adjusted constants.
QED
Remark 5.5.2 (Uniformity). The constants C_1, C_2 in (5.5.a) are
functions only of data fixed in advance: the curve C, the level
cover S_*, the center t_0, the frame, the connection, the chosen
ell, and the chosen height h. None depends on the varying
quaternion algebra D_t, its discriminant, or the endomorphism
order. This is what Section 6 requires to apply point counting.
The height Theorem 5.5.1 gives, for every QM point t of the
normalization C^nu (or of a fine cover S):
h(t) <= C_1 d(t)^{C_2}, (6.0)
with fixed constants independent of the target. This bounds the
height in terms of the degree. A further point-count argument,
due to Pila and Wilkie in the classical case (Pila and Wilkie) and
refined by Habegger-Pila (Habegger and Pila), then bounds the
degree itself. Northcott's theorem concludes.
The point-count argument has two ingredients:
(i) the definability, in the o-minimal structure R_{an, exp},
of the period map on the fine-level curve S(C);
(ii) the Habegger-Pila counting bound, applied to the rational
(flat) coordinates of the pair (v_1, v_2) at each target
QM point.
The argument occupies this section.
Fix the smooth projective completion Sbar of the fine-level curve
S, the ample height h on it, and the compact period quadric
Q_V subset P^4 (Section 1.6). Let Phi : S(C) -> Q_V be the period
map (open holomorphic map determined by the Hodge filtration).
Definition 6.1.1 (Flat chart). A chart will mean a simply
connected complex coordinate disk U subset S(C), equipped with a
flat trivialization of the integral local system V_Z on U (an
integral flat basis e_1, ..., e_5 of V_Z|_U). Given such a chart,
the period point [Phi(t)] and every integral endomorphism vector
v at a point t in U have flat-coordinate tuples in Q^5.
Lemma 6.1.2 (Counting coordinate bound). There is a finite
collection of charts (finite complex disks together with integral
flat bases) covering S(C), such that:
(a) The period maps Phi|_{chart} : U -> Q_V are definable in
R_{an, exp}.
(b) For every QM point t in S(Qbar), the integral pair (v_1, v_2)
at t (Lemma 4.1.2) has flat-coordinate tuple y in Z^{10}
satisfying
H_rat(y) <= c (d(t) h(t))^M (6.1.a)
in each chart containing t.
(c) The same bound holds for the conjugate pair at every
conjugate of t over K.
Proof. (a) Definition of charts. A compact subset of S(C)
away from the finitely many boundary points has a finite cover by
relatively compact simply connected coordinate disks. Shrink them
inside slightly larger disks where the flat trivialization of V_C
and the period map are holomorphic. Their real and imaginary
parts are restricted analytic functions, hence definable in
R_{an, exp}.
Near a puncture, choose u a local parameter on Sbar (algebraic
function with a simple zero at the puncture). The local monodromy
of V_Z is quasi-unipotent; on a finite cover u = w^e it is
unipotent, say exp(N) with N nilpotent. Schmid's nilpotent orbit
theorem [Schmid1973, Theorem 4.9] gives, in the compact dual, the
expression
Phi(u) = exp( (log w)/(2 pi i) N ) Psi(w), (6.1.b)
where Psi extends holomorphically across w = 0 with Psi(0) in the
compact dual. The exponential of a nilpotent matrix times log w
is a polynomial in log w; the branch of w is algebraic; |w| is
bounded away from 0 and 1; log|w| is definable in R_{an, exp}.
Restricting to a small punctured disk around u = 0 (with a fixed
branch of log w) gives a definable map. These punctured disks
supply finitely many further charts.
(b) Integral size of the pair. For the integral endomorphism
v = v_j at t, and for any complex embedding sigma of K(t), the
Weil operator on H^1 = H^{1, 0} (+) H^{0, 1} preserves the
weight-(0, 0) part. Hence the Hodge norm is
||v||_H^2 = <v, v> = (1/4) Tr(v_j v_j^dagger) = (1/4) q(v_j),
which by Lemma 4.1.2 is at most C (h d)^c. If v = sum_i n_i e_i
in the integral flat basis, then |n_i| = |e_i^*(v)| <= ||e_i^*||_H^vee *
||v||_H. Combining with the standard bounds on the dual Hodge
metric on the fixed finite chart gives
|n_i| <= c (d h)^{M/2} (6.1.c)
for every i and a fixed M. Hence H_rat(y) <= c (d h)^M.
(c) Conjugate pair. Galois conjugation preserves integrality
of v_1, v_2 and their q-norm q(v_j) = (1/4) Tr(v_j v_j^dagger).
Choosing the conjugate pair for the conjugate chart gives
(6.1.a) at every conjugate of t. QED
We use the following "path" form of the counting theorem, due to
Pila-Wilkie (Pila and Wilkie) and Habegger-Pila (Habegger and Pila).
Theorem 6.2.1 (Path counting; Habegger-Pila Cor. 7.2). Let
Z subset R^{15} be definable in R_{an, exp} and let
pi_1, pi_2 : R^{15} -> R^5 be the two coordinate projections.
Suppose Z carries a non-algebraic structure with the following
property: for every finite subset Sigma subset Z with
|pi_1(Sigma)| > c_Z(eta) T^eta
for some fixed eta > 0 and T exceeding the max of the
coordinate heights over Sigma, there exist
- a point z_0 in Sigma,
- a definable continuous path beta : (0, 1) -> Z with
beta(eps) -> z_0 as eps -> 0,
such that pi_1(beta) is nonconstant semialgebraic on (0, 1) and
pi_2(beta) is also nonconstant.
Here c_Z(eta) > 0 depends only on Z and eta.
Remark 6.2.2. We apply the theorem with Z = Z_chart (a definable
subset of R^5 x Z^{10}) and eta chosen small, as in Theorem 5.5.1
of [HabeggerPila2016, Sec. 7]. In the original paper
[05-counting.tex, Cor. 7.2] the same theorem appears.
Fix a chart U from Lemma 6.1.2. Let
Z_U := { (z_period, z_pair) : z_period in Q_V (subset C^5)
is the period point of a t in U,
z_pair in Z^{10} is the flat-coordinate tuple of
the pair (v_1, v_2) at that t,
the pair is integral, orthogonal to z_period,
and satisfies the Rosati-norm hypothesis }.
The set Z_U is definable in R_{an, exp}: the period image is
definable by Lemma 6.1.2(a); the integrality, orthogonality, and
Rosati-norm bound are algebraic conditions on z_pair; the pairing
is definable via the fixed flat basis.
Proposition 6.2.3 (Finite points with bounded height/degree).
The set of QM points t in S(Qbar) satisfying
h(t) <= A d(t)^{kappa} (6.2.a)
is finite, for any fixed A > 0 and kappa > 0.
Proof. Setup. Work in one chart U from Lemma 6.1.2, and consider
the definable set Z_U defined above.
Suppose for contradiction that the set of QM points t satisfying
(6.2.a) is infinite. For a target t, let sigma_1, ..., sigma_{r(t)}
be the distinct K-embeddings of K(t) into Qbar, r(t) = [K(t):K].
For each sigma_j, the conjugate pair (sigma_j(v_1), sigma_j(v_2))
is an integral pair at sigma_j(t) with the same Rosati norms. By
Lemma 6.1.2(c) it satisfies the flat-coordinate height bound
(6.1.a).
Count. Set T := c (d h)^M (from Lemma 6.1.2 with the particular
d, h from a fixed t and its conjugates). The conjugate points
Phi(sigma_j(t)) are r(t) distinct complex points on Q_V (via
Galois equivariance of Phi). Therefore #pi_2(Sigma) >= r(t).
For the applicablity of Theorem 6.2.1 what is needed is #pi_2(Sigma) >=
c_Z(eta) T^eta.
Choice of eta. Take eta > 0 small enough that eta M (1 + kappa)
< 1/2, i.e., eta < 1 / (2 M (1 + kappa)). Then (6.2.a) gives
T <= c (d A d^kappa)^M = c A^M d^{M(1 + kappa)}, so
T^eta < (c A^M)^{eta} d^{eta M (1 + kappa)} <= C d^{1/2}
for a fixed positive C = C(A, M, kappa).
Comparing: #pi_2(Sigma) >= r(t) = d(t)/J where J = [K:Q] is
fixed. For d(t) large, d(t)/J > C d(t)^{1/2}, i.e., (6.1.d)
holds, and Theorem 6.2.1 produces a definable path beta in Z_U.
Contradiction. Let beta = (z_period(s), z_pair(s)) be the path
from Theorem 6.2.1. By hypothesis pi_1 beta is nonconstant on
(0, 1) (so z_period(s) is a genuine complex curve in Q_V) and
pi_2 beta is nonconstant (so beta visits infinitely many distinct
period points). The pi_2 image contains infinitely many distinct
period points [Phi(t_s)] in S(C).
But pi_1 beta = (z_period(s))_{s in (0, 1)} is semialgebraic
(hence algebraic) subset of Q_V. Its complex Zariski closure has
dim C at most 1. Passing to the Zariski closure Z_period of the
whole image [Phi(t_s)] as s varies in (0, 1):
dim_C Z_period <= 1. (6.2.d)
But pi_2 beta nonconstant means there are infinitely many distinct
points Phi(t_s), so Z_period is a genuine algebraic curve (or
point), not Zariski dense in Q_V.
Contradiction with the Zariski density of Phi. By Proposition
2.1.1 (Hodge genericity), the image of the period map S(C) -> Q_V
is Zariski dense in the 3-dim quadric Q_V. Since S is open and
Z_period is a closed proper algebraic subset, contradiction.
Conclusion. The assumption that the set of QM points satisfying
(6.2.a) is infinite is false. QED
Combining Theorem 5.5.1 with Proposition 6.2.3 (A = C_1,
kappa = C_2) yields immediately:
Theorem 6.2.4 (Finite QM points on the fine-level curve).
The set of QM points t in S(Qbar) is finite.
Lemma 6.3.1 (Descent to coarse curve). The set of QM points in
C(Qbar) is finite.
Proof. The normalization C^nu -> C is finite; finiteness on
C^nu implies finiteness on C. The level cover S -> C^nu is
finite (Lemma finite-covers); by Theorem 6.2.4, S(Qbar) has
finitely many QM points. Their image in C^nu(Qbar) is finite; its
image in C(Qbar) (via normalization, finite) is finite. Adding back
the finitely many points discarded in Lemma finite-covers, we
conclude. QED
We now assemble everything into the proof of Theorem 0.5.1.
Theorem 7.1 (Main theorem; restated from Section 0.5). Let C be a
reduced irreducible closed Hodge-generic algebraic curve in A_2
over Qbar. Then the set
Sigma_QM(C) := { s in C(Qbar) :
End^0(A_s) is an indefinite quaternion
division algebra over Q }
is finite.
Proof. By Lemma finite-covers, choose a smooth connected finite-level
curve S with a finite surjective morphism S -> C and a principally
polarized abelian scheme A -> S whose coarse image is C. Enlarge the
coefficient field to a fixed K and fix an ample height h on the smooth
projective completion Sbar as in Section 5.
It suffices to show that S(Qbar) has finitely many QM points. Indeed,
the image of any finite subset of S(Qbar) is finite in C(Qbar), and
adding back the finitely many points discarded in Lemma finite-covers
preserves finiteness.
Now combine the two key bounds:
(i) Height bound (Theorem 5.5.1). For every QM point t of
S(Qbar),
h(t) <= C_1 d(t)^{C_2}, (7.1)
with constants C_1, C_2 fixed (depending only on C, the level
cover S_*, the center t_0, the frame, the connection, the
auxiliary prime ell, and the chosen height h).
(ii) Counting bound (Proposition 6.2.3 with A = C_1,
kappa = C_2). The set of QM points t in S(Qbar) satisfying
(7.1) is finite.
Applying (i) then (ii), the set of QM points in S(Qbar) is finite.
Therefore Sigma_QM(S) is finite. Its image under S -> C, together
with the finitely many points discarded in Lemma finite-covers, is
the full Sigma_QM(C). Hence Sigma_QM(C) is finite. QED
This proves Theorem 0.5.1.
The logical spine, on one page:
(1) OPERATOR MODEL (Section 1). On the fine-level curve S we
construct a rank-5 quadratic local system V of self-adjoint
trace-zero endomorphisms. At a QM point t, its integral
endomorphism plane E_t is a positive-definite 2-dim rational
sublattice spanned by actual integral endomorphisms. The
associated period quadric Q_V is a smooth 3-dim projective
variety inside P(V(x)C), and the period map Phi : S(C) -> Q_V
is holomorphic with holonomy in SO(V, q).
(2) MONODROMY AND ISOGENY FINITENESS (Section 2). The geometric
monodromy of V is Zariski dense in SO(V, q) (Proposition 2.1.1).
Between two Hodge-generic curves only finitely many polarized
isogeny correspondences exist (Proposition 2.3.2), giving a
fixed multiplier set M.
(3) INTERPOLATION ENGINE (Section 3). Theorem 3.2.1 states a general
theorem: given FIXED analytic data, a target has large height
only if its coordinates admit a certain polynomial vanishing that
strictly lowers the dimension of the ambient variety containing
it. The engine has two pieces: an effective algebra bound on
intersections, projections, and ideal membership in fixed
dimension (Section 3.4), and a uniform zero estimate at ordinary
points (Section 3.5), plus a graph/descent (Sections 3.6-3.8).
(4) ARITHMETIC INPUT (Section 4). Let B be the de Rham coordinates
of two fixed small integral endomorphisms at the centre t_0. Let
P be the de Rham coordinates of two small integral endomorphisms
at the target t (Lemma 4.1.2, Masser-Wustholz + bounded Galois
orbit). Proposition 4.2.1 bounds |P| by C(h + log d) globally,
and locally by C(1 + log h + log d) at prox places via a fixed
denominator-of-frame ideal J (Section 4.3). Proposition 4.5.2
builds, at every place in a fixed weighted set V, a common matrix
of trace relations pair{B}{Y(a)P} = T_v. Eliminating T_v via a
lifted Frobenius (Lemma 4.5.3) produces a single arithmetic
relation identical at every place in V. The local smallness sum
at V is bounded below in (4.5.i) by c h - O(1 + log^2(dh)).
(5) NON-IDENTITY (Section 5). The expression pair{B}{Y(a)P} - T
does NOT vanish identically along a = 0 on every branch W of
every subvariety containing z_*. Three cases:
(S1) constant system: monodromy SO(V, q) acts irreducibly,
so no nonzero B has pair{B}{Y(a)P} constant in
monodromy (Section 5.4.2);
(S2) paired system: same, using the two-argument monodromy
Sp_4 x Sp_4 (Section 5.4.3);
(S3) exceptional system: mod-ell auxiliary marking +
Frobenius quasi-isogeny, giving a signed action on the
two homogeneous endomorphism planes; unimodularity
rules out the negative sign, uNipotence the positive
(Section 5.4.4).
(6) HEIGHT BOUND (Theorem 5.5.1). Sections 3, 4, 5 combine: the
interpolation engine applied to the common-relation data
supplies, in contradiction to large h, a strict dimension drop.
Iterating to dimension 0 gives h(t) <= C_1 d(t)^{C_2}.
(7) COUNTING (Section 6). The period image in Q_V is definable in
R_{an, exp} (Lemma 6.1.2). The flat coordinate of the target
endomorphism pair is bounded by c (dh)^M. Pila-Wilkie /
Habegger-Pila: too many effective points force a definable path
with a nonconstant semialgebraic first projection in Q_V.
Zariski density of the period map on the 3-dim quadric is then
contradicted (Proposition 6.2.3).
(8) FINITENESS (Theorems 6.2.4, Lemma 6.3.1, Theorem 7.1).
Every step is unconditional: no boundary condition on C, no
bounded-reduction hypothesis on the target points, no bounded
discriminant of the target quaternion algebra.
Two features close the argument.
(a) Uniformity. No step of the argument reduces the target's
quaternion algebra, discriminant, or endomorphism order to a
bounded set before the height bound is established. This is what
distinguishes the argument here from earlier conditional finiteness
results ((Daw and Orr), (Daw and Orr), (Papas),
(Papas)). Section 4.1 uses a bounded Galois orbit plus
Masser-Wustholz to produce small integral endomorphisms at EVERY
target, without needing a bound on their Rosati norms.
(b) Non-effective in place. The height bound is effective in the
sense of giving a fixed polynomial threshold below which the
argument no longer needs to descend. It is NOT effective in the
sense of producing an explicit list of QM points on C; the finiteness
conclusion itself is ineffective (via the counting theorem), as is
usual for Zilber-Pink arguments.
(c) What could be relaxed. Hypothesis (i) of Theorem 3.2.1 requires
the target z_* to have a_* != 0 (i.e., not to lie on the divisor
a = 0). This is not a restriction on curves: the divisor a = 0 is
one boundary point of the fixed chart at the center t_0, and removing
it only excludes finitely many targets of S. Hypotheses (ii)-(iv)
are substantially the content of Sections 4 and 5.