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QUATERNIONIC DIVISION POINTS ON CURVES IN THE SIEGEL THREEFOLD
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A complete and self-contained account of the proof that a
Hodge-generic curve in A_2 over Qbar contains only finitely many
points whose abelian surface has division quaternionic
multiplication.
Abstract
--------
The Siegel threefold A_2 is the moduli space of principally
polarized abelian surfaces. It is an algebraic variety of
dimension 3. For each indefinite quaternion division algebra
D over Q, the locus of points whose abelian surface A has
End^0(A) = D (as algebras with positive involution) is a closed
algebraic curve S_D of dimension 1. The points of S_D are called
QM points of type D.
Zilber-Pink, in the precise form stated by Pink, predicts that a
Hodge-generic algebraic curve C in A_2 over Qbar contains only
finitely many points in the union of all S_D. The union is over
all division quaternion algebras at once, so the finiteness must
be uniform across all types of quaternionic multiplication, all
discriminants, and all maximal endomorphism orders.
This paper proves that prediction for the quaternionic-division
part. The main theorem (Theorem 0.5.1) is:
Every Hodge-generic reduced irreducible algebraic curve C
in A_2 over Qbar contains only finitely many QM points.
The theorem is unconditional. No restriction is placed on the
discriminant of the quaternion algebra, on the endomorphism
order, on the boundary of C, or on the reduction types of its
points.
The paper is written to be readable by a motivated reader with a
background in linear algebra and complex numbers, without prior
training in algebraic geometry. Every object we use is defined
at first appearance. Every external theorem we invoke is stated
and cited at the point of use.
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Contents
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0. Prelude: surfaces, endomorphisms, Zilber-Pink
1. The rank-five quadratic variation
2. Monodromy and isogeny correspondences
3. The interpolation engine
4. Arithmetic input: heights, local relations
5. Non-identity via an auxiliary marking
6. From heights to finiteness
7. Proof of the main theorem
References
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0. PRELUDE
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This section explains, without assuming background, the objects
that the theorem talks about and why the proof is nontrivial. A
reader who already knows what A_2 is and what a QM point is can
skip directly to Section 1.
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0.1 Abelian surfaces in one page.
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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.
An abelian surface is a complex torus of dimension 2 together
with a principal polarization, up to isomorphism. This is the
notion we are going to study.
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).
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0.2 Endomorphisms and endomorphism algebras.
------------------------------------------------------------------
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).
We 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.)
We will not need the full classification. What matters for us 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.
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0.3 Quaternionic-division points.
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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.
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0.4 The Siegel threefold A_2.
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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.
We 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.
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0.5 The main theorem.
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We can now state the result precisely.
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THEOREM 0.5.1 (Finiteness of QM points on curves).
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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.
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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.
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0.6 The Zilber-Pink prediction, in one page.
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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.
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0.7 What is known in the same direction.
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For a reference survey, see [DawOrr22], [Pap24], 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, [DawOrr22] reduce the proof to a "large Galois
orbit" hypothesis. Their later [DawOrr25] proves finiteness
against all special curves under the same boundary hypothesis.
[Pap24] extends to higher-dimensional A_g under degeneration
hypotheses; in dimension 2, the boundary assumption is the
same as in [DawOrr22].
[Pap25b] 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.
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0.8 Strategy of the proof, in outline.
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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. Let us outline.
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.
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END OF SECTION 0. Section 1 begins on the next turn.
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1. THE RANK-FIVE QUADRATIC VARIATION
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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.
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1.1 Setup: the fine-level curve S.
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Let C be the curve of Theorem 0.5.1. We freely pass 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 we obtain 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. We 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.
We will assume from now on, for the sake of contradiction:
S has infinitely many QM points. (Hyp-inf)
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1.2 The operator system V.
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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). We 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
------------------------------------------------------------------
1.3 The QM 2-plane E_t.
------------------------------------------------------------------
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 space we care about.
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), we have
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 we will write (P_1, P_2) for the de Rham coordinates of
(v_1, v_2), once we have set up the de Rham trivialization of V
in Section 1.5. The pair (B_1, B_2) will be the analogous pair at
the fixed centre t_0 (Section 4).
==================================================================
END OF SECTION 1.1-1.3. Sections 1.4-1.6 on the next turn.
==================================================================
------------------------------------------------------------------
1.4 The group-theoretic identification GSp_4 -> SO(V, q).
------------------------------------------------------------------
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 we must 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.
------------------------------------------------------------------
1.5 de Rham realization and isogeny conventions.
------------------------------------------------------------------
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".
------------------------------------------------------------------
1.6 The period quadric Q_V.
------------------------------------------------------------------
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.
* We will not need the finer analysis of the "primitive part"
as a GSp_4-module. The dimension count and the group map
GSp_4 -> SO_5 suffice.
==================================================================
END OF SECTION 1. Section 2 (monodromy + isogeny exceptions) next.
==================================================================
==================================================================
2. MONODROMY AND ISOGENY CORRESPONDENCES
==================================================================
The local information from Section 1 gives, at every QM point t,
a positive-definite 2-plane E_t inside V_t. When we will try
(Section 5) to prove non-identity of certain local equations, we
will need to know that monodromy "twists" those planes around
without invariant vectors. This section establishes the relevant
monodromy statements.
We also prove (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.
------------------------------------------------------------------
2.1 Single-argument monodromy.
------------------------------------------------------------------
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 [Deligne1979] 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 we will use 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).
------------------------------------------------------------------
2.2 Product monodromy: single product or isogeny.
------------------------------------------------------------------
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 we have 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, we conclude 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), we have 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.
==================================================================
END OF SECTION 2.1-2.2. Section 2.3-2.4 next; then Section 3.
==================================================================
------------------------------------------------------------------
2.3 Finitely many dominating isogeny correspondences.
------------------------------------------------------------------
We 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. We 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}. We 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). We 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), we need 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 we use 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.
------------------------------------------------------------------
2.4 Galois conjugates of the curve.
------------------------------------------------------------------
We will need 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.
==================================================================
END OF SECTION 2. Section 3 (interpolation engine) next.
==================================================================
==================================================================
3. THE INTERPOLATION ENGINE
==================================================================
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. We separate the statements ("what the
theorem says" and its lemmas) from the proof ("how we get 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.
------------------------------------------------------------------
3.1 Weighted absolute heights.
------------------------------------------------------------------
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 we 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.
------------------------------------------------------------------
3.2 The interpolation theorem: statement.
------------------------------------------------------------------
We need one more preliminary notion: "allowed analytic
expression".
Fixed analytic data. We 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. We 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.
We will descend 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', we get 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 3.4-3.8.
==================================================================
END OF SECTION 3.1-3.3. Sections 3.4-3.5 on the next turn.
==================================================================