ELLIPTIC SQUARES AND ZILBER-PINK FOR CURVES IN A_2
A self-contained proof with citations to the two companion articles
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HOW TO READ THIS
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This is the full proof of three theorems:
(E2) the elliptic-square case: a Hodge-generic algebraic curve
in the moduli space of principally polarized abelian surfaces
meets finitely many points isogenous to E x E, E without CM;
(ECM)/(Q) the two companion branches (E x CM and quaternion
division), proved in the two companion articles and quoted here;
(ZP) the full Zilber-Pink theorem for curves in A_2, obtained by
combining the three branches with Andre-Oort for points and
the classification of special curves.
The logical dependencies are:
(E2) proved here from scratch.
(Q) quoted from the companion article on quaternionic division.
(ECM) quoted from the companion article on E x CM curves.
(ZP) assembled here.
Every named lemma not proved in this text is either a classical theorem
with a precise bibliographic reference (Andre, Schmid, Deligne, Dube,
Aschenbrenner, Masser-Wustholz, Matthews-Vaserstein-Weisfeiler,
Berthelot-Ogus, Bertrand-Beukers, Moonen-Zarhin, Pila-Tsimerman,
Daw-Orr) or is one of the two companion theorems.
Notation conventions used throughout:
* All heights are absolute logarithmic heights; place weights
w_v = [L_v:Q_v] / [L:Q] sum to one and are preserved by
field extension.
* Constants C, kappa depend only on fixed data (the curve C, the
center, the auxiliary levels, the fixed frames). They never depend
on the varying target, its height, its degree, or the residue
characteristic.
* "Ordinary point" refers to a point where a linear connection has
no singularity; it is a property of a differential equation and
places no condition on the reduction of an abelian variety.
* Every reduction of an endomorphism plane uses the saturation of
the plane inside a specified integral lattice.
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PART 0. WHAT IS BEING MEASURED AND WHY
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0.1 Abelian surfaces up to isogeny.
An abelian surface over C is a complex torus C^2 / Lambda with a
Riemann form. Up to isogeny (a finite-to-one group homomorphism with
finite kernel) its isomorphism class is determined by the rational
Hodge structure H_1(A,Q), which is a Q-vector space of rank four
with a Hodge decomposition of type (-1,0)+(0,-1). The endomorphism
algebra End^0(A) = End(A) tensor Q is the algebra of all endomorphisms
of that rational Hodge structure.
0.2 Principal polarizations and the moduli space A_2.
A principal polarization is a symplectic form on Lambda, unimodular,
compatible with the complex structure. Principally polarized abelian
surfaces form a three-dimensional moduli space A_2 (coarse).
The moduli space is an orbifold: the fine moduli spaces A_2(N) for
N >= 3 are smooth quasiprojective varieties with a finite forgetful
map to A_2. Practically: to work on a curve C in A_2 we can always
pass to a fine cover, do the work there, and push the conclusion
back via finiteness of the cover.
0.3 Special subvarieties.
A Shimura subvariety of A_2 is called special (also: Shimura-special).
For our purposes it is an algebraic subvariety defined by the
requirement that the generic Mumford-Tate group be strictly smaller
than GSp_4. Special points are the CM points (the ones where
End^0(A) contains a commutative semisimple algebra of dimension 4,
equivalently a quartic CM field). Special curves in A_2 have three
generic types, classified by Moonen-Zarhin (Proposition 7.1 below):
(i) A is isogenous to E x F with E non-CM and F CM;
(ii) A is isogenous to E x E with E non-CM;
(iii)End^0(A) is an indefinite quaternion division algebra over Q.
Together with the Hecke translates (the translates by isogenies of
all possible degrees) these exhaust the special curves.
0.4 Zilber-Pink for curves in A_2.
For a closed irreducible curve C in A_2, let Sigma(C) be the set of
points of C(Qbar) that lie on a special subvariety Z of A_2 with
dim Z <= 1. The curve case of Zilber-Pink predicts: if C is Hodge
generic (contained in no proper special subvariety of A_2), then
Sigma(C) is finite. The three branches (i), (ii), (iii) of 0.3
split Sigma(C) into the special points already accounted for by
Andre-Oort and three curve-intersection problems, each of which must
be shown finite.
0.5 The three finiteness statements.
(E2) For every nonempty reduced irreducible Hodge-generic closed
curve C in A_2 over Qbar,
Sigma_{E^2}(C) := { s in C(Qbar) :
A_s is isogenous to E x E for some elliptic
curve E over Qbar without CM }
is finite. No condition on the closure of C at the
Baily-Borel boundary is imposed; no bound on the isogeny
degree or polarization is imposed.
(ECM) Same as (E2) but with E x F, E non-CM, F CM. Proved in the
E x CM companion article.
(Q) Same as (E2) but with End^0(A_s) an indefinite quaternion
division algebra. Proved in the companion article on
quaternionic division points.
Statement (E2) is what this article proves in full. Statements
(ECM) and (Q) are quoted. Combining all three with Andre-Oort for
points gives the full curve case (ZP).
0.6 Why branch (E2) is the hard one.
Branches (ECM) and (Q) admit arguments that use reduction theory in
a fairly direct way; branch (E2) is the place where the curve can
be arbitrarily close to the elliptic-square locus but not lying on
it, so one needs a quantitative handle that survives arbitrary bad
reduction and unbounded proximity to the special locus. That handle
is a polynomial height bound proved here (Theorem 7.4).
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PART 1. NOTATION AND THE INTERPOLATION TOOLKIT
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1.1 Heights.
For a point x = (x_1, ..., x_r) in affine space over a number field
F, define
h_aff(x) = sum_v w_v log max(1, |x_1|_v, ..., |x_r|_v),
w_v = [F_v:Q_v] / [F:Q].
The same symbol denotes the height of a polynomial's coefficient
tuple. For a smooth projective completion S of our curve, an ample
divisor D gives h_D; two ample divisors D1, D2 satisfy
h_{D1}(t) >= c h_{D2}(t) - O(1) for some c > 0, depending only on
D1, D2 and S. Throughout we use h = h_{[t_0]}(t) + c_0 with
c_0 a fixed constant so h >= 2; and d = max(2, [K(t):K]) with K a
fixed number field containing all fixed data.
1.2 Tensor convention.
On the rank-five space P of trace-free self-adjoint operators of
Part 3, write <D, E> = Tr(DE). Any two labelled pairs
D = (D_1, D_2), E = (E_1, E_2) in that space define a 2 x 2 matrix
<D, T E> = ( <D_i, T E_j> )_{1<=i,j<=2}
for any endomorphism T of the rational cohomology. Each matrix
equation below is really four scalar equations.
1.3 The interpolation theorem.
The proof of (E2) reduces the problem to a conditional
height bound for solutions of a small system of ordinary
differential equations at a finite set of p-adic (and archimedean)
places. This section states the theorem exactly; the proof is in
Part 2.
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PART 2. INTERPOLATION: STATEMENT AND PROOF
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2.1 Statement.
Fix: an ambient affine dimension N; a bound s_0 on the number of
equations; an integer b bounding the degrees of the polynomial
combinations used; a real constant C_0 >= 2; and a finite list of
"ordinary germs" - algebraic regular germs and ordinary horizontal
matrix germs (including dual and inverse matrices) on fixed algebraic
curves over number fields. From these data a constant C is determined.
Statement C is the following.
Let d, h >= 2 and let z = (a, U) in A^N(Qbar) with a != 0. Suppose
there is a weighted subcollection V of places and 1 <= s <= s_0
equations
E_alpha(a,U) = sum_{j>=0} a^j e_{alpha,j}(U) = 0, 1 <= alpha <= s,
satisfying the following four conditions.
(i) h_aff(z) <= C_0 d^{C_0} h, |a|_v < 1 for v in V,
sum_{v in V} w_v log |a|_v^{-1} >= h / (C_0 d^{C_0}),
sum_{v in V} w_v log^+ ||U||_v <= C_0 d^{C_0} (1 + log h)^{C_0}.
(The middle inequality is "proximity to the boundary a = 0";
the last says "the positive local logarithms of U remain small".)
(ii) The coefficients e_{alpha, j} are polynomials over a number
field containing the target and equation data with
deg e_{alpha, j} <= C_0 (1 + j),
h_aff(e_{alpha, j}) <= C_0 (1 + j + d)^{C_0} (1 + log h)^{C_0}.
The same equations are used at every selected embedding. They
converge at the target, and there are b_v >= 0 such that for
every ell >= 0 and every alpha,
| sum_{j >= ell} a^j e_{alpha, j}(U) |_v
<= |a|_v^ell * exp((ell + 1) b_v),
sum_{v in V} w_v b_v <= C_0 d^{C_0} (1 + log h)^{C_0}.
(iii) As analytic functions in (a, U), each E_alpha is a polynomial
combination of bounded degree (<= b) of the coordinates and
the fixed germs evaluated at a (and possibly at aR, where R is
one of the coordinates of U). Coefficients may depend on the
target with the above bounds. Every expression converges near
any finite point of a = 0.
(iv) Let W be any irreducible closed subvariety of A^N over Qbar
containing z on which a is nonconstant. For every prime divisor
D on the finite normalization of W lying over a component of
W cap {a = 0}, at least one E_alpha is not the zero germ on
the local branch at a general point of D.
Then h <= C d^C.
Note that C depends only on the fixed list (N, s_0, b, C_0, germs,
curves). It does not depend on d, h, or the target.
2.2 Why the graph construction lowers boundary dimension.
Choose an irreducible algebraic subvariety W of A^N containing z
with controlled degree and defining height (initially W = A^N).
Write k = dim W and suppose a varies on W. Hypothesis (iv), together
with the ordinary differential multiplicity estimate to be recalled,
gives an integral linear combination E of the E_alpha and an integer m
such that the truncation F = E_{<m} := sum_{j<m} a^j e_j(U) satisfies
ord_D F < m ord_D a
at every normalized boundary divisor D. This strict inequality means
the rational function g = F / a^m blows up along every boundary
divisor of W.
Let W' be the closure of the graph of q = g over W cap {a != 0}.
Its projection pi : W' -> W has generic degree one and dim W' = k.
Over the boundary a = 0, g has a pole along every boundary divisor,
so W' has no point over the generic point of any boundary divisor of
W. Consequently pi(W' cap {a = 0}) has dimension <= k-2. Imposing the
extra equation q + e_m(U) = 0 pins down the graph coordinate, so
dim ( W' cap {a = 0, q + e_m(U) = 0} ) <= k-2,
and if k = 1 the intersection is empty.
The point of this reduction: the auxiliary interpolation problem for
small degree polynomials is now solved on a (k-2)-dimensional
boundary instead of a (k-1)-dimensional one. Saving one dimension in
the boundary is exactly the factor that lets us run a product-formula
argument with interpolation data small relative to degree.
2.3 The lift of the target.
Lift z to z' = (a, U, q(z)) on W' where q(z) = a^{-m} F(z). Because
E vanishes at z, q(z) = -a^{-m} sum_{j >= m} a^j e_j(U). By the tail
bound (ii), the positive local logarithm of this new coordinate is
bounded by (m+1) b_v at each v in V, at worst. So z' still has "small
positive local logarithms" at selected places, even though its global
height is roughly proportional to h.
2.4 The polynomial P.
Choose equations generating the ideal I of W'. Consider the augmented
ideal I_n = (I, a^n, F_n) where
F_n = q + sum_{j=0}^{n-1} a^j e_{m+j}(U).
Lemma (Layered interpolation, below): for every prescribed integer
c > 0 there is a polynomial P of degree <= L, nonzero on W', with
P in (I, a^n, F_n), n = c L,
where L is polynomial in (c plus the degree and height of the defining
data) but independent of h.
The point of the exact ideal rather than its radical: if P actually
belongs to the ideal, then a certificate of membership can be
substituted and evaluated. This is what makes "P in the ideal" and
"P vanishes on W'" interchangeable in the right direction, which we
need at the selected places.
2.5 The product formula.
At v in V the equation E(z) = 0 lets us rewrite the certificate in
the form "P(z') = a^n * (unit)". So |P(z')|_v gains a factor |a|_v^n.
Even if the certificate has degree larger than L, its degree
multiplies only the small positive local logarithms b_v.
At v not in V we evaluate P itself (degree L), so the global cost is
at most L times the global height of z'. Combining those estimates:
if P(z') != 0, then
0 <= ( - n / A(d) + L C(d) ) h + D(d) (1 + log h)^B,
where A, C, D are fixed polynomials, B is fixed, and A, C do not
depend on c. Increasing c (polynomially bounded in d) makes the
coefficient of h negative, forcing h <= C' d^{C'}.
if P(z') = 0, then substituting q = F / a^m and clearing a-power
denominators gives a polynomial H(a,U) = a^{mL} P(a, U, F(a,U)/a^m)
that is nonzero on W (because P is nonzero on the graph closure)
but zero at z (because a(z) != 0). So z lies on the proper
subvariety W cap {H = 0}, which has strictly smaller dimension and
controlled height. Condition (iv) applies to that subvariety again,
so we iterate.
After at most N dimension drops, either h is bounded polynomially in
d or we reach W disjoint from {a = 0}. In the second case, if
F_1, ..., F_r define W set-theoretically then (F_1, ..., F_r, a) = (1)
(an ideal with empty zero set is the unit ideal), so there is an
identity 1 = a A + sum rho F_rho A_rho with polynomial degrees and
heights controlled in the same way. At z we get 1 = a A(z), and the
ordinary size inequality for a polynomial evaluated at z gives
1 <= |a A(z)|_v <= exp(D_* (d)(1 + log h)^{B_*} )
combined with the proximity sum; the h on the right absorbs via
(1 + log h)^B <= c_B h^{1/2}, giving h <= C d^C. That special case
covers every subvariety on which a is constant and every
zero-dimensional W.
So the whole proof reduces to three concrete lemmas:
(L1) the effective algebra bounds (Groebner, membership certificate,
set-theoretic equations),
(L2) the ordinary differential multiplicity bound,
(L3) layered interpolation in the actual ideal.
We now state and prove each.
2.6 Lemma L1 (effective algebra, Dube + Aschenbrenner + Krick-Pardo-Sombra).
Fix the number of variables M. Let D >= 2 bound a collection of at
most D polynomials in M variables, each of degree <= D and coefficient
height <= H. Then:
(a) The ideal has a degree-compatible Groebner basis with
polynomially bounded degrees and coefficient heights <=
poly_M(D)(1 + H). Each basis element has a representation in
the original generators with the same kinds of bounds, and a
linear normal form reduction exists whose extra degree is
Delta <= poly_M(D).
(b) Any polynomial of degree T in the ideal has a certificate of
membership with degree <= poly_M(D + T) and coefficient height
<= poly_M(D + T) (1 + height of input).
(c) Every irreducible component of the zero set has degree
poly_M(D), set-theoretic equations of degree poly_M(D) and
defining height <= poly_M(D)(1 + H).
These are absolute bounds; enlarging the number field changes nothing.
Proof sketch. (a) is Dube's Groebner degree bound plus an explicit
linear-algebra height deduction, spelled out in the appendix of the
original article: the monomial coefficient matrix of every identity of
prescribed degree has poly_M(D) entries, each of height <= H, and
Cramer's rule for a nonzero pivot minor produces a solution of height
poly_M(D)(1 + H) with normalized absolute heights (no factor
[F:Q] appears, because heights are already normalized by [F:Q]).
(b) is Aschenbrenner's effective Nullstellensatz with the same
determinantal argument. (c) follows components by proper hypersurface
cuts (at most M cuts, geometric Bezout for degree) and the global
arithmetic intersection bound of Krick-Pardo-Sombra for height.
The precise numerical form is in the references
[Dube, SIAM J. Comput. 1990; Aschenbrenner, JAMS 2004;
Krick-Pardo-Sombra, Duke 2001].
What matters for the main proof is only the *polynomial character*
of the bounds: every operation performed on a variety increases
degree and defining height by a polynomial function of the input
degree and height. Tracked once at the start, this remains true
through every dimension drop.
2.7 Lemma L2 (ordinary differential multiplicity, Bertrand-Beukers
with a varying-curve upgrade).
There is a polynomial M_0(D) with the following property. Let
W subset A^N be irreducible of degree <= D with a nonconstant, and
suppose normalized boundary branches satisfy condition (iv). Then
there are integers 0 <= c_alpha <= M_0(D) and an integer 1 <= m <= M_0(D)
such that with E = sum c_alpha E_alpha and its truncation F = E_{<m},
every prime divisor D_0 of the normalization over W cap {a = 0}
satisfies ord_{D_0} F < m ord_{D_0} a.
The constants depend only on the fixed function data, not on W's
defining height or on the target-dependent coefficients.
Proof sketch.
1. Slice W by k-1 = dim W - 1 general affine hyperplanes to
obtain smooth curve germs transverse to D_0 at a general point.
Choose the hyperplanes generically, and also to avoid the zero
loci of "first nonzero transverse coefficients" of all E_alpha
(analytically this is possible; the closed condition is
algebraic, using that the leading transverse coefficient is an
algebraic section of a power of the conormal bundle).
2. On the resulting smooth algebraic curve X (a component of the
sliced W with the fixed-germ coordinates adjoined), every
E_alpha is of the form r_0(x) y(x) for a fixed linear
connection and a fixed row r_0. Taking successive d/da
derivatives: rows r_j satisfy a recurrence r_{j+1} = delta r_j
+ r_j B, where B is the connection matrix. This is exactly the
Bertrand-Beukers derivative-span argument.
3. The rows r_0, ..., r_{s'} for s' <= fixed rank become linearly
dependent at the first s', and the span is stable under delta.
Smith normal form over the local ring O = C{t} after clearing
poles gives a primitive vector: some entry of some r_j is a
unit. Hence the minimum of ord_0(delta^j f) over 0 <= j < s'
is bounded by (2 s' - 1) P where P bounds the polar degree of
all r_j and of B. The P and s' depend only on D (through the
degree of X and of a, R, and the fixed row/connection
functions) — not on the coefficients of E_alpha.
4. If e = ord_0 a >= 1 and n_0 = ord_0 f, differentiation gives
ord_0(delta^j f) >= n_0 - j e; combining with step 3 bounds
n_0 / e polynomially in D. Pulling back by the finite
normalization multiplies both orders by the same ramification
index, preserving the ratio.
5. Averaging over all divisors D_0: pick, for each divisor, the
least order among the nonzero E_alpha. Cancellation at that
order in sum c_alpha E_alpha is a proper hyperplane condition
on (c_1, ..., c_s). There are at most D divisors, hence at most
D forbidden hyperplanes. A hyperplane cannot contain the whole
integer grid {0, ..., D}^s (fix all coordinates but one whose
coefficient is nonzero). Choose (c_1, ..., c_s) outside all
forbidden hyperplanes.
6. Choose m larger than every ratio (n_0 / e) of step 4. Any
coordinate function of W is regular on D_0, so
E - F = a^m * (analytic function); its order is >= m ord a.
Hence ord F = ord E < m ord a. Done.
The constants M_0(D) depend only on the fixed function data and N,
not on the coefficients of the individual equations or the target.
2.8 Lemma L3 (layered interpolation, in the actual ideal).
Fix M. Let I = (I_1, ..., I_r) be generated by polynomials of
controlled number, degree, height. Suppose its zero set W' is
irreducible of dimension k >= 1. Set
f_0 = q + e_m(U), J = (I, a, f_0),
F_n = f_0 + sum_{j=1}^{n-1} a^j e_{m+j}(U),
dim V(J) <= k - 2 (emptiness when k = 1),
deg e_{m+j} <= C_0 (1 + m + j).
Let D >= 2 bound r, m, the generator degrees, and C_0. There are
B, B' polynomial in D (for fixed M) such that whenever
B' n (1 + L + Bn + B)^{k-2} < binom(L + k, k)
there is a polynomial P with deg P <= L, P nonzero on W', and
P in (I, a^n, F_n).
For k = 1 the left side is replaced by zero and any L >= 1 suffices.
Both P and a certificate for the membership have degree polynomial
in (D + L + n) and coefficient height polynomial in (D + L + n)
times (1 + input coefficient heights).
Proof sketch.
The proof is a linear algebra count.
1. Fix a degree-compatible Groebner basis of J and reductions
of basis elements to bounded-degree combinations of
I_1,...,I_r,a,f_0 (Lemma L1). This gives a fixed linear
"normal form" map NF_J and a linear division identity
D_1 = NF_J(D_1) + sum I_rho A_rho(D_1) + a A(D_1) + f_0 H(D_1),
with the property: if deg D_1 <= T then every coefficient
polynomial on the right has degree <= T + Delta, where
Delta = poly_M(D) is independent of T, and NF_J(D_1) also
has degree <= T.
2. Start with a general P of degree <= L. Layer 0 residual: D_0 = P.
Layer i (0 <= i < n) residual D_i. Impose NF_J(D_i) = 0 and
push the "a-multiple remainder" to layer i+1, and using
a^i f_0 H(D_i) = a^i F_n H(D_i) - sum_{j>=1} a^{i+j} e_{m+j}(U) H(D_i),
push the "e_{m+j} H(D_i)" remainders into specific layers.
After the last layer there is no residual.
The layer-by-layer identity gives
P in (I, a^n, F_n).
3. Every layer's residual degree grows by at most B j when
advanced by j layers, so deg D_i <= L + B i. The number
of independent linear conditions is at most
B' n (1 + L + Bn + B)^{k-2}:
count monomials in ker NF_J of degree <= T, using the
initial-ideal structure: if r_0 = dim V(J) <= k - 2, the
space of normal forms of degree <= T has dimension
<= B' (1 + T)^{r_0}. Summing over layers 0, ..., n-1 and
using deg D_i <= L + B i gives the stated count.
4. Restrictions of degree-<=L polynomials to W' have dimension
>= binom(L + k, k) (k independent linear coordinates on W'
suffice; monomials in them of degree <= L are independent).
So if the condition count is smaller, some P of degree <= L
satisfies all layer conditions and is nonzero on W'.
5. Height control: the membership P = sum B_rho I_rho + B_a a^n +
B_f F_n is a homogeneous linear system with polynomially many
unknowns of degree polynomial in D + L + n; Lemma L1(a) gives
a basis of solutions whose coefficient heights are polynomial
in D + L + n times (1 + input height). At least one basis
vector has P nonzero on W' (otherwise no solution would
have P nonzero on W', contradicting step 4).
6. Choosing n = c L: the condition count is at most
B' c (2 + B c + B)^{k-2} L^{k-1}, and binom(L + k, k) >=
L^k / k!. Taking L > k! B' c (2 + B c + B)^{k-2} suffices;
this L is polynomial in D + c.
The exact ideal (not radical, not taking powers of P) is preserved
through the entire layered construction. That is why the certificate
evaluates to a multiple of a^n at selected places.
2.9 Conclusion of Theorem 2.1.
Combining L1-L3 with the descent argument of 2.5 concludes the
interpolation theorem. The dimension drops are bounded by N; every
step has polynomial degree and height bounds in (d, input height);
constants absorb the finitely many choices. There are exactly two
endgame possibilities:
* the descent stops with h bounded polynomially in d, or
* W becomes disjoint from {a = 0} and the Nullstellensatz identity
1 = a A + sum F_rho A_rho bounds h polynomially in d.
Either way h <= C d^C.
This finishes the interpolation theorem. Everything else in the
paper is verification that a specific arithmetic problem satisfies
the numerical input conditions (i)-(iii) and the nonidentity
condition (iv).
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PART 3. THE FIVE-DIMENSIONAL SELF-ADJOINT SYSTEM AND MONODROMY
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3.1 Working on a fine cover.
Fix a Hodge-generic closed curve C in A_2 over Qbar, reduced and
irreducible. Suppose the required elliptic-square locus is infinite.
Choose distinct primes r_1, r_2 >= 3 and N = r_1 r_2. There is a
fine moduli space A_2(N) of principally polarized abelian surfaces
with full symplectic level N: an abelian scheme over the smooth
quasiprojective A_2(N), with a finite surjective forgetful map to A_2.
Pull back to C, normalize, remove the preimage of any finite set. We
work with:
* a smooth geometrically irreducible curve S over a fixed number
field K_0,
* a principally polarized abelian scheme A -> S,
* a finite map S -> C onto a dense open,
* two full symplectic levels at r_1 and r_2,
* a fixed center t_0 in S(Qbar) that lies in the elliptic-square
locus.
Degree comparison. After enlarging the fixed number field K to
contain all these data, there is a constant b >= 1 (fixed) with
[K(s):K] <= [K(t):K] <= b [K(s):K]
for any lift t in S(Qbar) of a retained point s of C. Reason: t lies
in a fiber of the fixed finite cover S -> C, fiber lengths are fixed,
and residue field degree of a fiber point is at most the fiber length.
Subsequent finite covers (markings below, removal of finite sets)
change b by a bounded factor.
Fixed conjugates. Applying embeddings of the field K_C defining C
gives finitely many coarse curves C_1, ..., C_e in A_2. All conjugate
families below have coarse images in this list. Enlarging K to define
the center or the markings does not add coarse images: the image of C
depends only on the restriction of an embedding to K_C. We always
keep the full coarse curves C_1, ..., C_e, even if the working
parameter curves get restricted to open subsets.
3.2 The trace-free self-adjoint system.
Let H = R^1 pi_* Q, with pi : A -> S the structure map of the abelian
scheme: rational covariant first homology of the fibers, a rank-4
local system of Q-vector spaces over S. The polarization gives an
alternating pairing psi : H x H -> Q(1) with the standard Tate twist
in the relevant realizations. For an endomorphism D of H, denote by
D^dagger its psi-adjoint: psi(Dv, w) = psi(v, D^dagger w). Define
P = { D in End(H) : D^dagger = D, Tr_H(D) = 0 },
<D, E> = Tr_H(DE).
All realization functors (de Rham, etale prime-to-characteristic,
Betti) commute with this construction.
Proposition 3.1 (five-dimensional orthogonal system).
P is a rank-five rational vector space with a split nondegenerate
symmetric trace pairing. Conjugation gives a surjection
GSp_4 -> SO(P)
with scalar kernel; restricting to Sp_4 gives the spin covering
Sp_4 -> SO(P),
kernel {1, -1}. P is an absolutely irreducible representation of
SO(P). For every odd prime ell, the integral lattice
Lambda_ell = { D in End_{Z_ell}(T_ell A) : D^dagger = D, Tr D = 0 }
is free of rank five with unimodular trace pairing.
Finally, let A_t be isogenous to E x E with E non-CM (over Qbar). Then
F_t = End^0(A_t) cap P_t
is a rational two-dimensional plane inside P_t on which the trace
pairing is positive definite.
Proof.
Choose a symplectic basis of H over Q with Gram matrix
J = (0 I_2 ; -I_2 0).
Self-adjointness is exactly D^t J = J D. Writing D in 2x2 blocks,
the two conditions D^t J = J D and Tr D = 0 give
D = (A u K_2 ;
v K_2 A^t),
A = (a b ; c -a),
K_2 = (0 1 ; -1 0),
parametrized by (a, b, c, u, v). Then
D^2 = (a^2 + b c - u v) I_4,
Tr_H(D^2) = 4 (a^2 + b c - u v).
The trace form in coordinates (a, b, c, u, v) has Gram matrix
diag(2, 0, 0, 0, 0) plus hyperbolic cross terms (a, b), (a, c),
with determinant 64. So the form is split (signature (3, 2) over R)
and unimodular over Z_ell for every odd ell. That proves the
integrality claim and unimodularity.
GSp_4 is connected; its image in O(P) therefore has determinant 1,
so lies in SO(P). The image is nontrivial because conjugating the
matrices above gives nontrivial orthogonal transformations. The
kernel is a normal subgroup of GSp_4 contained in the center; the
smallest nontrivial such subgroup is the {1, -1} inside Sp_4, which
acts trivially on P (a scalar lambda acts by lambda^2 * lambda^{-2}
= 1 on trace-free self-adjoint parts modulo the polarization bijection).
At the Lie-algebra level, dim sp_4 = 10 = dim so_5, and the kernel on
sp_4 is zero, so the image is all of so_5. Consequently Sp_4 -> SO_5
is a covering; Sp_4 is simply connected and SO_5 is not, so it is the
double cover (spin covering) with kernel {1, -1}.
Irreducibility of SO(P): the natural SO_5 representation V_5 is
irreducible over any field of characteristic zero. Its exterior power
Lambda^2 V_5 = so_5 decomposes as so_5 itself (adjoint) plus one
scalar; the trace-free self-adjoint part is exactly the primitive
exterior square of Sp_4's standard four-dimensional representation,
which is irreducible. Equivalently: the SO_5-orbit of any nonzero
vector (isotropic or nonisotropic) spans V_5.
Positivity. Write A = A_t = E x E (up to isogeny) with E non-CM.
Then End^0(E) = Q (since E is non-CM), and
End^0(A) = End_{Hdg}(V) tensor End(W) = M_2(Q),
where V = H_1(E, Q) and W = Q^2 with trivial Hodge structure, so
H_1(A, Q) = V tensor W. Choose a polarization psi_E on V. Every
rational Hodge alternating pairing V x V -> Q(1) is a rational
multiple of psi_E; applying this to the four matrix entries of the
polarization psi_s of A shows psi_s = psi_E tensor b for a symmetric
rational bilinear form b on W. Positivity of psi_s and psi_E then
forces b positive definite over R.
Take the positive square root of b to reduce everything to
b-orthonormal coordinates. In these coordinates, any psi_s-self-
adjoint trace-free D in F_t is of the form 1 tensor D_0 with D_0 a
symmetric traceless 2x2 real matrix. Trace-free means tr(D_0) = 0;
self-adjoint means D_0 symmetric. For D_0 != 0,
Tr_{H_1(A)}(D^2) = Tr_{V tensor W}((1 tensor D_0)^2)
= Tr_V(1) * tr_W(D_0^2) = 2 tr_W(D_0^2) > 0.
That is the positivity on F_t.
Crucially, the proof uses only that A is isogenous to E x E and is
given a principal polarization; it does NOT require the isogeny to
preserve the polarization. The polarization enters only through b,
and b is obtained directly from psi_s by composing with psi_E.
3.3 The fixed integral pair at the center.
Fix a labelled integral pair B = (B_1, B_2) of linearly independent
integral trace-free self-adjoint endomorphisms spanning F_{t_0}.
(Clear denominators in any rational spanning pair.) Similarly, at
any target t, construct a labelled integral pair P = (P_1, P_2)
spanning F_t, whose sizes are bounded in Part 5.
Saturated plane lattices: for any non-CM elliptic-square target u,
let
Pi_{u, ell} = (F_u tensor Q_ell) cap Lambda_ell,
a saturated rank-two direct summand of Lambda_ell. Its image mod ell
is an embedded two-space in Lambda_ell / ell Lambda_ell. A full
integral symplectic ell-marking identifies the ambient rank-five
space with a fixed V_{F_ell}. We never identify the reduction of a
chosen pair (B_1, B_2) or (P_1, P_2) with the reduction of Pi: a
chosen pair need not be an integral basis of its plane. The
reductions always use the saturated Pi.
At the center, the restriction of the trace pairing to Pi_{t_0, ell}
is nondegenerate mod ell for all but finitely many ell: exclude the
prime divisors of the Gram determinant of an integral Betti lattice
basis of F_{t_0}. Betti-Tate comparison gives precisely the lattices
above.
3.4 Connected monodromy.
Proposition 3.2 (monodromy).
The connected algebraic monodromy of H is Sp_4, and of P is SO_5.
This remains true on every fixed finite cover, on a dense open, and
for the conjugate families under consideration. It is also true for
the pullback of the local system to a smooth connected algebraic
variety mapping dominantly to the base curve.
Suppose W is a smooth connected algebraic variety mapping dominantly
to two such curves S_1, S_2. Then either the connected joint monodromy
on the two copies of P is SO_5 x SO_5, or the coarse joint image of W
lies in a polarized isogeny correspondence.
Proof.
Hodge genericity in A_2: the generic Mumford-Tate group of H is
GSp_4. The variation is pure and polarizable and comes from an
abelian scheme over a smooth algebraic curve. Andre's normality
theorem (S5 Thm 1) makes connected monodromy a normal subgroup of the
derived generic Mumford-Tate group = Sp_4. Since Sp_4 is almost
simple, connected monodromy is trivial or all of Sp_4.
Trivial is excluded. If connected monodromy were trivial, a finite
cover would trivialize the integral local system; the period map
would be single-valued holomorphic into Siegel space H_2 (a
three-dimensional bounded domain in C^3). Each coordinate of that
map is bounded near the finitely many missing points of the smooth
projective completion, hence extends, hence is constant on the
compactification. So the family would have constant moduli image,
contrary to dominating C. So the connected monodromy is all of Sp_4.
Passage to P: Sp_4 -> SO(P) surjective with finite kernel => connected
monodromy of P is SO_5.
Invariance under finite covers or open subsets: a finite etale cover
of an open of the curve has fundamental group of finite index; finite-
index subgroups have the same Zariski closure. Dominant pullback:
choose a general curve inside the higher-dimensional algebraic variety
where the projection to the base is nonconstant; the pullback to that
curve is already full; higher-dimensional W contains its monodromy.
Conjugation: Betti Zariski density propagates to etale Zariski density
via comparison; field conjugation identifies geometric etale families
and their symplectic Tate systems; so conjugates inherit full Betti
monodromy.
Joint monodromy. Work at a generic Mumford-Tate point of the direct
sum of the two pulled-back homology local systems, in rational
symplectic bases. Both projections of connected monodromy are full
(by the curve-case-cited-above argument). Pass to the adjoint groups
G_1, G_2, each isomorphic to PGSp_4 = SO_5. Let M subset G_1 x G_2 be
the connected joint image. The kernel of either projection is a normal
subgroup of the other adjoint simple group; hence either one kernel
is a whole factor (giving M = G_1 x G_2), or both are trivial and M
is a graph
M = { (x, phi(x)) : x in G_1 }
for a Q-isomorphism phi : G_1 -> G_2. This is the group form of
Goursat.
Lift the adjoint graph to a rational similitude. Let T be the generic
Mumford-Tate group of the direct sum. Normality on W says T normalizes
M. The normalizer of the graph in G_1 x G_2 is the graph itself
(trivial center of G_2), so T is contained in the graph. Root diagram
C_2 has no nontrivial automorphisms, so phi is inner:
phi(x) = g x g^{-1} for a rational g in GSp_4. Then the resulting
scalar character c between the two Hodge homomorphisms (a scalar
character of the Deligne torus) satisfies c^2 = 1: both Hodge
homomorphisms hit the polarization character Q(1), and the projection
to the adjoint is the same. The Deligne torus is connected, so c = 1.
Positivity of the similitude multiplier nu(g): apply the polarization
formula psi_2(gv, J_2 gv) = nu(g) psi_1(v, J_1 v), which forces
nu(g) > 0. An integral multiple of g is therefore a polarized
isogeny.
Consequently, if W's joint image is not an adjoint product, it lies in
the coarse Hecke correspondence T_g for some g in GSp_4(Q)^+. The
correspondence T_g is closed and algebraic (image of a moduli space of
polarized isogenies with multiplier nu(g).)
This proves the proposition.
3.5 Finitely many dominating Hecke types.
A dominating relation between coarse curves C_i, C_j is an
irreducible algebraic curve in (C_i x C_j) cap T_g with both
projections dominant.
Proposition 3.3 (finite Hecke list).
For the fixed collection C_1, ..., C_e of conjugate coarse curves,
only finitely many coarse Hecke correspondences admit a dominating
relation between a pair of them. Hence there is a fixed finite list
Tsym containing every exceptional joint image from Proposition 3.2.
The list Tsym can be assumed closed under exchanging the two factors.
Additionally, for each T_g in Tsym we can fix a positive integer
multiplier m_g and a fixed finite set of rational simplitudes
realising every pair on T_g: f_g : A_t -> A_s with f_g^* lambda' =
m_g lambda. The list Tsym and the multipliers are fixed before choosing
any additional auxiliary levels.
Proof. Use one fixed torsion-free principal level N as in 3.1. The
inverse images D_alpha of C_i in A_2(N)(C) are finitely many
irreducible closed curves, each with full connected monodromy by 3.4.
Write G = GSp_4(R)^+ / R^x, isomorphic to Sp_4(R) / {1, -1}. Let
Gamma_N be the level-N arithmetic group. The uniformization
pi_N : H_2 -> A_2(N)(C) is a covering (because the level is torsion-
free).
I. Lift components and their integral monodromy groups.
Remove the finitely many singular points of D_alpha and pick a
connected component U_alpha of its inverse image. Let Y_alpha be the
closure of U_alpha in H_2: a closed irreducible analytic curve.
This is by local analytic normality; the closure joins precisely its
branch continuations, and the normalization of the closure is
connected.
Put Delta_alpha = Gamma_N cap Stab_G(Y_alpha). Covering-space monodromy
identifies this with the image of pi_1(D_alpha^reg) in Gamma_N.
Zariski density of Delta_alpha in G follows from 3.4 (which identifies
the connected monodromy already at the level of pi_1).
II. L_alpha = Stab_G(Y_alpha) is closed and discrete.
Closedness: g_n in L_alpha, g_n -> g gives g Y_alpha subset Y_alpha
by closedness of Y_alpha; apply to inverses for equality. So L_alpha
is a closed subgroup of the real Lie group G.
Lie algebra: it is Ad(Delta_alpha)-invariant. The stabilizer of a
linear subspace of sp_4(R) under the adjoint action is algebraic;
Zariski density therefore makes Lie(L_alpha) invariant under all of
G, hence an ideal in the simple Lie algebra sp_4(R). So Lie(L_alpha)
= 0 or all. All would imply L_alpha = G (connected), which would
make the nonempty curve Y_alpha invariant under the transitive action
of G on H_2, impossible. So Lie(L_alpha) = 0 and L_alpha is discrete.
III. Area of Delta_alpha backslash Y_alpha^reg finite.
Equip H_2 with its invariant Hermitian metric. On a broken-off
punctured disk 0 < |q| < R in the normalization of Y_alpha (= the
coarse D_alpha^reg), the local Schwarz lemma near a point gives a
uniform metric bound: moving the value at 0 to the standard point
iI_2 in H_2 via an element of G, and using the Cayley transform to a
bounded domain U of symmetric 2x2 matrices of operator norm < 1,
the scalar Schwarz applied to u^* f v for unit vectors gives
||f'(0)||_{op} <= 1, which at the origin is a multiple of
tr(dZ dZ^*); pulling back by disk automorphisms gives the metric bound
f^* ds^2_{H_2} <= c / (1 - |z|^2)^2 dz dbarz
for every holomorphic disk map f and absolute constant c.
At a puncture q -> 0 on the coarse curve, lifting the universal cover
to H_2 gives
ds^2 <= c' |dq|^2 / (|q|^2 log^2(R/|q|)).
Its area integral over 0 < |q| < r < R gives c' times
integral_0^r dho / (rho log^2(R/rho)) = 1/log(R/r), finite.
Compact remainder has finite area by smoothness. Hence the claimed
finite area.
IV. [L_alpha : Delta_alpha] < infty.
The G action on H_2 is proper: H_2 = G/K with K compact. Its
restriction to closed discrete L_alpha is proper, with finite point
stabilizers. Choose y in Y_alpha^reg and F = (L_alpha)_y finite.
Properness gives a small neighborhood U of y in Y_alpha^reg with
hU cap U = empty unless h in F.
For representatives h_i of distinct double cosets in
Delta_alpha backslash L_alpha / F, the images of h_i U in
Delta_alpha backslash Y_alpha are disjoint (an intersection forces
h_i^{-1} delta h_j in F for some delta in Delta_alpha, i.e. same
double coset). Each image has area >= Area(U)/|F|. Infinitely many
distinct double cosets would give infinite total area, contradicting
III. Finite double cosets + finite F gives finite index.
V. Transporters and rational double classes.
For two lift components, the transporter
R_{alpha, beta} = {g in G : g Y_alpha = Y_beta}
is either empty or a right coset of L_alpha. By IV it has finitely
many right Delta_alpha-cosets. Among those that meet GSp_4(Q)^+,
choose rational representatives; each class contributes finitely many
integral double classes of the full GSp_4(Q)^+.
Every dominating relation gives one of these classes:
Over a smooth point of a dominating component of (C_i x C_j) cap T_g,
lift local arcs to the torsion-free level and to H_2. A local sheet of
T_g identifies the two lifted arcs by a rational positive similitude
in the integral double class of g. The arcs live in some translates of
Y_alpha, Y_beta; apply deck transformations to bring them to those
components. The translated similitude takes a nonempty open arc of
Y_alpha into Y_beta. Analytic identity principle makes g Y_alpha =
Y_beta. So g lies in the transporter R_{alpha, beta}. Finitely many
pairs (alpha, beta) times finitely many double classes per pair gives
finiteness of the list.
VI. Integral multipliers.
For each retained rational double class choose a rational
representative; multiply by a positive integer to clear denominators;
the multiplier m_g = nu(g) is then an integral positive integer (by
integrality of the symplectic form: applying g to a pair of integral
basis vectors with psi(u, v) = 1 shows nu(g) is an integer). Then g
realizes a polarized isogeny f_g with f_g^* lambda' = m_g lambda on
every pair of T_g. Repeat for the inverse directions. Finitely many
integers.
Finitely many such positive integers are produced; they are fixed
before any auxiliary markings at ell_1 (Proposition 4.2) are chosen.
That closes the geometric section.
=====================================================================
PART 4. AUXILIARY LEVELS AND BAD-CENTER EXCLUSION
=====================================================================
4.1 Finite field linear algebra.
Lemma 4.1 (inc:finite-fields).
There is an absolute constant q_0 such that for every finite field k
of odd cardinality q > q_0, every nondegenerate plane H_0 subset V_k
(the trace-free self-adjoint part of a symplectic four-space over k,
dimension 5 with the quadratic trace pairing), and every plane
H subset V_k (rank of restricted trace form 0, 1, or 2), there exist
g_+, g_- in Sp_4(k) with
dim (H_0 + g_+ H) = dim (H_0 + g_- H) = 4,
and with the H_0 + g_- H sum degenerate, the H_0 + g_+ H sum
nondegenerate.
The action is the quadratic representation V_k induced from the
natural Sp_4(k) action on a symplectic four-space. We do NOT need
surjectivity of Sp_4(k) -> SO(V_k); the conclusions are about the
specific generating group Sp_4(k).
Proof.
Step A. Nondegenerate sum.
Work first over kbar. Embed H_0 inside a nondegenerate four-space
M = H_0 perp K with dim K = 2. Choose bases in which H_0 and K carry
the standard form I_2. A graph from K to H_0 with matrix
A = diag(a, b) is transverse to H_0, and its restricted form is
I_2 + A^t A = diag(1 + a^2, 1 + b^2).
By choosing a, b in kbar we realize every rank in {0, 1, 2}. Bilinear
forms on a 2-space over kbar are classified by rank, so there is a
graph isometric to H (including when H is degenerate). Witt extension
extends the isometry to an orthogonal transformation O(V_kbar). Change
determinant by a reflection in a nonisotropic vector perpendicular
to the graph (that perpendicular is 3-dim, while a maximal totally
isotropic 2-space exists), giving an element of SO(V_kbar). Spin
cover gives an element of Sp_4(kbar).
Step B. Degenerate sum.
Choose an isotropic line L subset H_0^perp over kbar. The
perpendicular 3-space is nondegenerate isotropic: after diagonalization
the sets {a x^2 : x in kbar} and {-c - b y^2 : y in kbar} each have
(q+1)/2 elements and therefore meet for odd q. So L exists.
The group Sp_4(kbar) is transitive on isotropic lines of V_kbar.
Indeed, under the primitive exterior square, isotropic lines are
Plucker lines of Lagrangian 2-spaces in the symplectic four-space;
symplectic bases show transitivity.
Move the chosen line in H^perp to L. After this preliminary move,
H subset L^perp. Let P subset Sp_4 be the stabilizer of L; put
Q = L^perp / L, a nondegenerate ternary space. Over kbar, the Levi of
P acts on Q by all of SO(Q).
Case B1: H contains L. Its projection is a line in Q. Move outside
that of H_0 (standard rep of SO(Q) irreducible). Then H cap H_0 = 0.
Case B2: H does not contain L. Its projection is a plane H in Q. Move
H so it does not coincide with the projection of H_0 (a plane cannot
be invariant under the irreducible SO(Q)). The two projected planes
meet in a line J. The unipotent radical of P permits arbitrary change
of lift Q -> L^perp by Q -> L. Explicitly, with hyperbolic pair e, f,
L = kbar * e, <e, f> = 1, Q = (kbar e + kbar f)^perp, the
orthogonal unipotent is
e |-> e, n |-> n - <n, z> e, f |-> f + z - (1/2) <z, z> e.
These lift into the unipotent radical; equivalently, in the symplectic
realization, the radical consists of matrices (I Z; 0 I) with Z
symmetric. Choose the linear form so the nonzero-vector lifts from
the moved H and the fixed H_0 are distinct; then H cap H_0 = 0.
In either case, the concatenated bases give H_0 + g_- H,
four-dimensional because g_- H is a plane in L^perp disjoint from H_0
and both in a four-space L^perp, and their sum is in L^perp. Its
radical is L; four-dimensional inside a nondegenerate five; radical
1-dim and equal L; hence degenerate.
Step C. Rational point via algebraic charts.
Both Step A and Step B are "open nonempty" conditions on g of
bounded degree over k. Write these conditions on a big Bruhat cell of
Sp_4:
A^4 x G_m^2 x A^4,
multiplying negative root subgroups, torus, positive root subgroups,
in a fixed order. The matrix, its inverse, and its action on V_k are
Laurent polynomials of bounded degree in those coordinates
(independently of k). A big cell of a split parabolic (Levi root
subgroups + unipotent radical) behaves the same way. Conjugating a
parabolic or inserting one fixed element changes coefficients but not
these degree bounds.
Hence an open condition of bounded degree that holds at some kbar-point
of the chart is given by a nonzero polynomial in bounded-many
variables and bounded degree. A nonzero polynomial of total degree D
in n variables cannot vanish on all of k^n once #k > D (induct on n,
use the univariate root bound). So there is a k-point once q exceeds
a q_0 that depends only on the fixed dimension and root system.
Conclude: for q > q_0 the required g_-, g_+ exist in Sp_4(k).
4.2 One connected cover carrying both auxiliary markings.
Proposition 4.2 (inc:markings).
After one fixed finite etale cover pi' : S' -> S of the fixed smooth
curve S, there are distinct fixed odd primes ell_1, ell_2 and a fixed
lift t'_0 of t_0 with the following properties. Every non-CM
elliptic-square target of C has a lift t' in S'(Qbar) such that, in
the respective marked finite-field spaces,
Pi_{t'_0, ell_1} + Pi_{t', ell_1} is a degenerate four-space,
Pi_{t'_0, ell_2} + Pi_{t', ell_2} is a nondegenerate four-space.
Moreover:
* ell_1 divides no fixed Hecke multiplier from 3.5;
* ell_2 is different from every residue characteristic at which the
fixed center is not potentially good;
* both center-plane reductions are nondegenerate;
* conditions survive simultaneous conjugation of all the data;
* degree of any chosen lift in S' over its coarse image point in C
grows by at most a fixed factor.
Proof.
Let Gamma subset Sp_4(Z) be the geometric monodromy group of the
integral symplectic Betti lattice along the fixed curve S. Gamma is
finitely generated (fundamental group of a smooth algebraic curve)
and Zariski dense (Proposition 3.2).
Strong approximation for Gamma (Matthews-Vaserstein-Weisfeiler):
Gamma -> Sp_4(F_ell) surjective for ell outside a finite set.
Fix ell_1 > q_0 (from 4.1), outside the MVW exceptional set, outside
the primes dividing fixed Hecke multipliers from 3.5, and outside the
finitely many ell at which the restriction of the trace pairing to
Pi_{t_0, ell} is degenerate. The kernel Gamma_1 of
Gamma -> Sp_4(F_{ell_1}) has finite index. Schreier: Gamma_1 is
finitely generated. Its Zariski closure is still Sp_4 (a connected
algebraic group is not a finite union of translates of a proper closed
subgroup).
Strong approximation applies to Gamma_1; choose ell_2 > q_0 distinct
from ell_1, outside the new exceptional set, outside the finitely many
ell at which Pi_{t_0, ell} is degenerate into the trace form, and
outside the finitely many residue characteristics at which A_{t_0} is
not potentially good (finite because fixed abelian variety has good
reduction away from finitely many primes).
Conclusion:
Gamma -> Sp_4(F_{ell_1}) x Sp_4(F_{ell_2}) surjective.
(Realize any prescribed element in the first factor; adjust its second
factor using an element of Gamma_1.)
The cover of S parametrizing pair of full symplectic markings with
fixed Weil pairings is connected because its monodromy acts
transitively on the product of marking torsors. After a fixed
extension of the field of definition it is a fixed geometrically
connected finite etale cover S' -> S.
For each target coarse point s of C: apply 4.1 to the pair
(Pi_{t_0, ell_1}, Pi_{s, ell_1}) with ambient marked spaces, to find
an ell_1-marking yielding the degenerate sum; apply 4.1 independently
to (Pi_{t_0, ell_2}, Pi_{s, ell_2}) to find an ell_2-marking yielding
the nondegenerate sum. Connectivity of the marking cover means both
markings are simultaneously realized in one lift t' in S'(Qbar).
Selection need not be an algebraic rule: every lift in the fiber has
residue field degree over s bounded by the fiber length, which is
the fixed degree of S' -> S.
Subsequent finite omissions in later sections only omit finitely
many points downstairs, cf. 3.1.
Simultaneous conjugation: for gamma in GSp_4(Qbar), the map
D |-> gamma D gamma^{-1} is an orthogonal transformation on P up to a
scalar on the trace pairing. Scalar multipliers do not affect
degeneracy or nondegeneracy of four-space sums. Both incidences
survive. Potential good reduction is invariant under finite extension,
so adjoining levels does not introduce new non-potentially-good
residue characteristics.
4.3 A common toric subspace in a strict local model.
Lemma 4.3 (inc:torus).
Let X be a regular integral model of the Siegel space near the
Baily-Borel boundary, with G -> X a smooth semiabelian family of
relative dimension 2 with geometrically integral fibers. Suppose the
interior carries a full ell-marking, ell invertible on X. Let zbar be
a geometric point whose fiber is a two-dimensional torus. Then:
* any two traits through zbar whose generic points lie in the
interior give the same marked two-space W_zbar subset (Z/ell Z)^4
coming from the special torus torsion;
* along each trait it lifts to a saturated rank-2 lattice T subset
T_ell A;
* every actual integral endomorphism of the generic abelian surface,
after extension of its field of definition, preserves T;
* and if F subset End^0(A) is a rational two-plane, the reduction
of (F tensor Q_ell) cap Lambda_ell also preserves W_zbar.
Proof.
Take R = O^{sh}_{X, zbar}, the strict henselization of the local ring
at zbar. This is a noetherian regular local ring, hence an integral
domain. The inverse image U of the moduli interior in Spec R is a
nonempty open subset of an integral scheme, hence is connected.
Finite-part decomposition of G[ell^n].
Multiplication by ell^n on the smooth semiabelian scheme is etale
(differential is multiplication by invertible ell^n). Its kernel
G[ell^n] is separated etale quasi-finite over R. The finite-part
decomposition over a henselian local base splits G[ell^n] into a
finite etale open-and-closed piece E_n and a piece with empty closed
fiber. Over R, since the closed fiber is the full torus torsion
((Z/ell^n Z)^2 with cardinal ell^{2n}), the closed fiber is finite of
order ell^{2n}; hence E_n has etale rank ell^{2n} and the "spectral"
piece is empty. So G[ell^n] = E_n.
E_n is finite etale over the strictly henselian R, hence constant;
each of its closed points has a unique lift to an R-section. The
multiplication and transition maps between E_n and E_{n+1} are the
unique lifts of the standard ones. Hence the system
(G[ell^n])_{n >= 1} is isomorphic to the standard (Z/ell^n Z)^2
system after choosing compatible generators in the special torus.
This is the standard finite-part construction of quasi-finite flat
subgroups (Lan, S3.4.1, equations (3.4.1.1)-(3.4.1.4)).
Toric labels. Over U, the full ell-marking identifies G[ell]_U with
the constant group (Z/ell Z)^4. Every section of E_1 has a constant
label under this identification (U connected). The labels on the
two-dimensional special torus form a 2-dim subspace W_zbar subset
(Z/ell Z)^4.
For any trait through zbar (generic in interior), lift to strict
henselization, and use compatibility of finite parts under local base
change; the generic toric torsion of that trait has exactly these
labels. Hence every trait sees the same W_zbar.
Lifting to a saturated lattice T subset T_ell A. Along a fixed trait,
take inverse limit over n of the generic fibers of E_n. This yields
a rank-2 lattice T subset T_ell A whose mod-ell reduction is the
embedded W_zbar. Saturation: lift a basis of W_zbar to compatible
generators of the system E_n; the resulting two Tate vectors have
Z_ell-independent mod-ell reductions in the free rank-4 Tate module;
extend to a Z_ell-basis; hence their span is a direct summand. Its
inverse limit is precisely T. So T is saturated.
Endomorphism action. An integral endomorphism f of the generic
abelian surface extends to the semiabelian scheme on the trait by the
normal-base extension theorem for semiabelian schemes
(Faltings-Chai I, Prop 2.7; Lan Prop 3.3.1.5). On the special fiber
f preserves the torus and its ell^n-torsion; unique-lift of torsion
sections gives preservation of every E_n, hence of T. Rational
combinations preserve T_{Q_ell}. Since T is saturated,
T = T_{Q_ell} cap T_ell A.
Every integral endomorphism of the generic abelian surface that lies
in the rational span preserves this intersection. In particular,
(F tensor Q_ell) cap Lambda_ell preserves T. Reducing mod ell gives
preservation of W_zbar.
Note: argument uses only henselian-local facts and prime-to-
characteristic torsion; no comparison between de Rham horizontal
transport and bad-reduction limit is used.
4.4 Stabilizer lemma.
Lemma 4.4 (inc:stabilizer).
Let V_4 be a symplectic four-space over a field of characteristic not
dividing 2. No nondegenerate four-space of trace-free self-adjoint
endomorphisms of V_4 can preserve a two-dimensional subspace W of V_4.
Proof.
Case 1: psi|_W has rank 2. Then W has symplectic orthogonal W^perp of
rank 2 with V = W perp W^perp. A psi-self-adjoint D preserving W
preserves W^perp, hence is diagonal: D = a I_W + b I_{W^perp}. Trace-
free gives 2a + 2b = 0, so (a, b) = (a, -a). Stabilizer of W inside
trace-free self-adjoint endomorphisms is 1-dimensional. A four-space
of such operators cannot fit inside a 1-dimensional subspace.
Case 2: W is Lagrangian (psi|_W = 0). Choose W as the first Lagrangian
in a symplectic basis so that the form matrix is (0 I; -I 0).
Trace-free self-adjoint endomorphisms preserving W are
( D J ;
0 D^t )
with Tr D = 0 and J^t = -J (the latter is 1-dimensional), giving
a 4-space (dim sl_2 = 3 for D, plus 1 for J). A direct computation
gives X(D, J) X(D', J') = (D D', DJ' + JD'^t; 0, D^t D'^t), so
Tr(X(D, J) X(D', J')) = 2 tr(D D').
The J-slot does not appear in the pairing, so the entire
1-dimensional J-part {(0, J) : J skew} is in the radical. On the
D-slot the pairing restricts to the trace form on sl_2, which is
nondegenerate in odd characteristic. Hence rad(stabilizer) is
exactly the J-part, the whole four-space stabilizer is degenerate,
and any four-subspace of it is degenerate.
Conclude: no nondegenerate four-space of trace-free self-adjoint
operators preserves a 2-subspace.
4.5 Bad-center disk exclusion.
Proposition 4.5 (inc:bad-disks).
Fix a finite place v of the field K of the center t_0 at which A_{t_0}
is not potentially good. There is a radius r > 0, fixed by the center
and finitely many fixed data, such that no selected target from the
marking cover of Proposition 4.2 lies in the actual-branch disk
{ t in S'(Qbar) : |x(t)|_v < r, t -> t_0 in the branch }
of the completed local curve at v. The radius is independent of the
finite, possibly ramified extension over which a target is defined.
The same conclusion holds for every conjugate of the fixed data.
Proof.
Local extension. After a fixed finite local extension (which does not
change any conclusion), the elliptic curve E with A_{t_0} ~ E x E
has semistable reduction: pick a fine full elliptic level invertible
at v and apply the proper toroidal construction in dimension 1.
An elliptic curve with semistable reduction is good or multiplicative.
Toric rank is invariant under isogeny (push isogeny and quasi-inverse
to semiabelian models; on tori the two composites are multiplication
by an integer, hence isogenies) and additive on products. So the
potential toric rank of A_{t_0} = E x E is zero or two. Rank zero in
the semistable model is good reduction, which would make A_{t_0}
potentially good, contrary to hypothesis. So the potential toric rank
is two.
Toroidal setup at ell_2. Use the full ell_2-level toroidal model X,
which is available at this residue characteristic (Proposition 4.2
chose ell_2 not equal to any non-potentially-good residue char). By
properness, the center point t_0 extends to a section over the
valuation ring, after a fixed local extension. Its pullback
semiabelian scheme is the semistable model of A_{t_0} (uniqueness of
semibelian extension on a normal trait). So the geometric special
fiber is a two-dimensional torus. Denote the specialization in the
toroidal model by zbar.
Disk argument. Choose a small affine open U subset X containing zbar.
Enlarge the field so that U has center coordinates defined. The
generic image of t_0 is in U; pull U-coordinate generators f_j back
to rational functions on the curve that are regular near t_0. Their
values at t_0 are integral. Pick a local parameter x at t_0 with
x(t_0) = 0. Continuity gives r > 0 with
|x(t)| < r => |f_j(t) - f_j(t_0)|_v < 1
for every j. Use the branch of x through t_0. Each f_j has a
convergent power series expansion on a fixed smaller disk, so the
inequalities hold in the fixed algebraic closure of the local field,
for points in every finite extension, and involve no varying
ramification index. The integral values f_j(t) satisfy the chart
equations and have the center's residues, so define an integral
section with special point zbar; separatedness identifies the generic
point with the given moduli point.
Contradiction from a bad target. If a selected target t lay in this
disk, then the two traits (through t_0 and t) would fall under
Lemma 4.3 with ell = ell_2. Both Pi_{t_0, ell_2} and Pi_{t, ell_2}
would preserve the same W_zbar subset (Z/ell_2 Z)^4 in their marked
symplectic four-spaces. Their sum is a nondegenerate four-space by
Proposition 4.2. But Lemma 4.4 shows a nondegenerate four-space of
trace-free self-adjoint operators cannot preserve a fixed 2-space.
Contradiction. So the target lies strictly outside the disk.
Finitely many non-potentially-good places, finitely many conjugate
data; simultaneous conjugation preserves the incidence 4.2; get
finitely many radii. Done.
Note. The argument uses only integral models and prime-to-characteristic
torsion. It provides what is needed for the good-reduction
comparisons of Part 6 without touching a bad-center de Rham
comparison.
=====================================================================
PART 5. HEIGHTS AND INTEGRAL FRAMES
=====================================================================
5.1 Fixed height convention.
Retain the fixed data: S, the abelian scheme A -> S, the center
t_0 in S(Kbar), the labeled pair B = (B_1, B_2) spanning
F_{t_0} = End^0(A_{t_0}) cap P_{t_0},
the auxiliary levels ell_1, ell_2 from 4.2, and the fine cover
pi' : S' -> S over which both markings are simultaneously realized.
For a varying target t in S'(Kbar), write
K = the fixed number field containing all fixed data,
d = max{2, [K(t):K]},
h = h_{[t_0]}(t) + c_0.
Here h_{[t_0]} is the absolute logarithmic Weil height on the smooth
projective completion Sbar with [t_0] as a fixed ample divisor, and
c_0 is a fixed shift making h >= 2 always. Since Sbar is projective
and [t_0] is ample, h is comparable with any fixed Weil height up to
a fixed additive constant. The shift c_0 absorbs that difference.
Notice: h is deliberately one height, and all height bounds below
are of the form h <= C d^C in terms of this same h. This is what
makes the "if h is small, done; otherwise derive a bound" logic of
the interpolation theorem legal.
5.2 Small endomorphisms.
Proposition 5.1 (ht:endomorphisms).
There is C > 0 fixed, and there are kappa > 0 fixed, such that for
every t in S'(Qbar) satisfying
- A_t is isogenous to E^2 with E non-CM,
the following all hold. Put Q_t(f) = Tr_{H_1(A_t)}(f f^dagger).
(1) h_F(A_t) <= C h (Faltings height).
(2) |disc End(A_t)| <= C (d h)^kappa, where the discriminant is
the discriminant of the full order End(A_t) with respect to
the trace of the regular representation of M_2(Q).
(3) There exist integral endomorphisms f_1, ..., f_4 in End(A_t)
spanning the full endomorphism algebra End(A_t) tensor Q,
and a labeled integral trace-free self-adjoint pair P_1, P_2
in End(A_t) spanning the self-adjoint trace-free plane
F_t = End^0(A_t) cap P_t, all with
Q_t(f_i), Q_t(P_j) <= C (d h)^kappa.
(4) All these endomorphisms, and hence their de Rham coordinates,
are defined over an extension L / K(t) with [L : K(t)]
bounded by a constant depending only on n = 2 and the fixed
data.
Proof.
(1) After a fixed finite theta-level cover S_theta -> S (independent
of t), a symmetric ample line bundle with theta data gives an
algebraic theta-null map into projective space. On S_theta, use the
pullback of [t_0] as ample divisor; the theta-null height at a lift
of t is h + O(1) by the height comparison in 3.1. Hence the theta
height is at most C h. Pazuki's comparison of theta height with
stable Faltings height (Bull. SMF 2012, Cor 1.3(1)) gives
h_F(A_t) <= C h. The logarithmic error in Pazuki's comparison is
absorbed into the constant. The theta cover has fixed degree, so it
does not bound d from above.
(2)/(3) Masser-Wustholz (Math. Z. 1994, Proposition S4), with the
induction parameter m = n = 2 and a principal polarization, gives
integral endomorphisms spanning End^0(A_t) with Rosati quadratic
sizes at most
c max{2, [L:Q], h_F(A_t)}^{kappa_2},
where c and kappa_2 depend only on n = 2. The dependence on
[L:Q] is polynomial and appears in (S5, eq (5.3)) and (S5, Lemma 5.1)
of that reference; it is not an unspecified constant allowed to
depend arbitrarily on [L:Q]. From the spanning set f_1, ..., f_4:
* the trace-free self-adjoint part is spanned by (f_i + f_i^dagger)
modulo the identity component (each has Rosati size at most
4 Q_t(f_i)); and
* clearing denominators of a rational basis gives the integral
trace-free self-adjoint pair P_1, P_2 within a bounded integer
multiplier.
Only the bound on the sizes needs verification; the spanning
property is algebraic. Both f_i and P_j then satisfy
Q_t <= C (d h)^kappa, with exponents absorbed into kappa.
(4) The field of definition statement is the Geometric
Endomorphism Algebra theorem of Masser-Wustholz (same section,
S5). It says that the coefficients of an integral basis of End(A_t)
in any fixed rational frame are algebraic over K(t), with degree
bounded by a constant depending only on n = 2 and the fixed data.
The trace-free self-adjoint pair is then obtained by a fixed
rational linear combination. Enlarging to the compositum of the
fields of the f_i and P_j gives a single L / K(t) of bounded degree.
5.3 Integral vector extension.
We need a functorial integral lattice whose formation commutes with
finite base change and whose endomorphism action is integral. The
model is the canonical extension of the de Rham cohomology of a
semiabelian family to the boundary.
Proposition 5.2 (ht:integral-model).
Let T be a regular noetherian integral scheme, U subset T a dense
open, and G -> T a smooth separated semiabelian scheme of relative
dimension 2 with geometrically integral fibers. Suppose G_U is an
abelian scheme with a principal polarization. Then:
(a) The Poincare bundle on G_U x_U G_U has a unique extension as a
biextension B on G x_T G, normalized (trivialized at the
simultaneous origin and with the two generic axis
rigidifications uniquely extended).
(b) B defines a smooth vector-group extension
0 -> omega_{G/T} -> E -> G -> 0,
omega_{G/T} = e^* Omega^1_{G/T}, whose Lie algebra H = Lie(E)
is a locally free O_T-module of rank 4. On the characteristic-
zero abelian open, H identifies functorially with
H_1^dR(G_U/U).
(c) For any trait Spec R -> T whose generic point lies in U, and
any integral endomorphism f of the generic abelian variety,
the induced action of f on H_1^dR preserves the integral
lattice H_R. Preservation remains after every finite extension
of the trait on which f is defined.
Proof of (a).
Start with any extension B_0 of the Poincare bundle from G_U^2 to
G^2 = G x_T G. Write p : G^2 -> T for the structure map. Replace B_0
by
B = B_0 tensor p^*((e, e)^* B_0)^{-1}.
This normalization makes the fiber of B over the simultaneous
origin trivial (as a line bundle restricted to the section (e, e)).
Rigidifications. On a biextension, one specifies line trivializations
along the two axis rigidifications (identity at the origin). The
generic (over U) trivializations extend uniquely to T: the divisor of
any rational trivialization is pulled back from T, and restriction to
the origin, where it equals the identity, forces that divisor zero.
A rational section of a line bundle on a regular scheme with zero
divisor is everywhere invertible.
Biextension law. Consider the associativity identity on G^3. It is a
condition on the ratio line bundle
R_3 = (m, 1)^* B tensor p_13^* B^{-1} tensor p_23^* B^{-1},
which is a line bundle on G^3 with a rational trivialization. Its
vertical divisor is detected at the triple origin (e, e, e), where
the trivialization equals the identity, so the divisor is zero.
Hence the rational trivialization is a nonsingular section.
Compatibility of the two laws and unit identities: same argument,
checked at the origin.
Uniqueness of the normalized extension: same argument, quotient of
two candidates gives a line bundle on G^2 with trivialization equal
at the origin.
This is the divisorial argument used by Edixhoven-Lido (JIMJ 2023,
Prop 6.7.1) over a trait; the above generalizes to arbitrary
regular T.
Proof of (b).
Over a test scheme V -> T, an E-section is a G-section g plus a
trivialization of B restricted to {g} x N_V, where N_V is the
first infinitesimal neighborhood of the identity in the second G-
copy, extending the specified value at the origin. Since N_V has
square-zero ideal sheaf omega_{G/T} tensor O_V, sections of E (as
bundles) exist locally on V; any two differ by an element of
1 + omega_{G/T} tensor O_V (square-zero, so this is additive).
Concretely: with q : G x_T N -> G the finite flat projection and
pi : G -> T the structure map, axis normalization gives an exact
sequence of vector bundles
0 -> pi^* omega_{G/T} -> q_*(B|_{G x_T N}) -> O_G -> 0.
The inverse image of the section 1 is the affine E-torsor. Its
points are actual sections (no quotient by automorphisms of
semiabelian extensions). The first biextension law supplies the
group law, and the fiber at the identity is the vector group
omega_{G/T}. Smoothness of the whole extension E follows from
smoothness on this affine chart, and O_G-linearity from the
component-wise diagram. Taking Lie algebras gives a sequence
0 -> omega -> H -> Lie G -> 0
of bundles of ranks 2, 4, 2. Identification with H_1^dR is the
standard canonical extension: on the characteristic-zero abelian
open, H is isomorphic to the coherent sheaf of sections whose
de Rham class extends horizontally; this is the canonical
extension of the Gauss-Manin connection to the boundary.
Proof of (c).
An integral endomorphism f of the generic abelian surface
functorially acts on the Poincare bundle, hence on the normalized
biextension B on G_U^2. The extension B on G^2 is unique, so f
extends to a biextension automorphism.
At the level of E-sections, the induced action on H = Lie(E) is
integral: an E-section over an integral scheme whose generic fiber
comes from f's action extends by uniqueness of the normalized
biextension. This identifies the f-action on H with the endomorphism
of the whole E-bundle structure, and in particular it preserves the
integral structure H_R over any trait R -> T.
Finite extension of the trait: the extension of f is unique (again
by the biextension normalization), and preservation of H_R survives.
No integrality assumption on the polarization of the trait, and no
Neron-model comparison, is used.
5.4 Coordinates and the near-frame estimate.
Fix a rational frame of P_{dR}. Write [F] for the five coordinates
of an element of P_{dR} in that frame. Fix affine coordinates
u_1, ..., u_b on a fixed chart of S containing t_0 and germs f_nu
with u_nu(t) = f_nu(a) on its parameter disk. Write P for the labeled
trace-free self-adjoint pair (P_1, P_2) attached to a target; we
often conflate the pair with the 2 x 5 matrix of its coordinates.
Proposition 5.3 (ht:coordinates).
There is C > 0 fixed such that:
(1) Global affine height:
h_aff(P) <= C (h + log d).
(2) Local proximity sum: let V be any subcollection of places in
the actual center branches, with bad-center places excluded as
in 4.5. Then
sum_{v in V} w_v log max(1, |P_1|_{v,coord},
|P_2|_{v,coord})
<= C (1 + log h + log d).
(3) For a fixed rational positive integer N and any fixed finite
set of endomorphisms F_j with N F_j integral, the analogous
estimates for the F_j also hold, with Q_t(F_j) <= C(dh)^kappa.
(4) If f : A_t -> A_{t'} is a polarized isogeny of one of the
fixed multipliers (from 3.5), write I_t = Ad(f) on P_dR.
Then the estimates hold for the transported pair
P^ = I_t^{-1} P', with P' the pair at A_{t'}, and with the
added field degree from choosing f bounded by a fixed constant.
Proof of (1).
Choose an algebraic theta-level cover just as in 5.2. The endomorphism
rings f_1, ..., f_4, P_1, P_2 are defined over L / K(t) with
[L : K(t)] bounded. Height comparison for projective varieties gives
h_aff([F_j]) <= C h_K for a fixed projective-height comparison; the
coordinate height of P_j and F_j in the fixed rational frame is at
most a fixed multiple of h_K. Combined with the E^2 origin of the
target (5.2) and the K-degree bound [L:K], we get
h_aff(F_j) <= C(h + log d). The pair P is a fixed rational combination
of the F_j with bounded coefficients, so the same bound holds.
Proof of (2): archimedean and finite separately.
Archimedean.
Equip H_1(A_t, C) with the Hodge metric defined by the principal
polarization. For an actual endomorphism f, Rosati adjunction is
metric adjunction, hence
|f|_{op,Hdg} <= Q_t(f)^{1/2}.
Fix an algebraic frame of P_{dR}. The map sending an endomorphism to
its de Rham matrix [f] in that frame is a fixed algebraic map, whose
matrix entries have projective degree bounded by C h from (1). The
operator norm and the frame coordinates are related by fixed frame
constants as long as the frames remain invertible. At archimedean
punctures of S, Schmid's Hodge norm estimates (Invent. Math. 1973,
Thm 6.6') give that the frame norms and dual norms grow at most as
C |u|^{-M} (1 + |log |u||)^M
near each missing point; finitely many sectors suffice, and the bound
is independent of the target.
Meromorphic coordinates from endomorphisms to P_{dR} have the same
type of bound. After taking positive logarithms and summing at
archimedean places, the total contribution is at most a fixed
multiple of a local height of the puncture plus a constant, which via
effective divisor comparison is at most C h. On a fixed small disk
about t_0 these frame norms are simply bounded.
Finite places: two-regime argument.
On the fixed integral models T_i = Spec O_K[1/ell_i], i = 1, 2, the
integral vector extension H_i of 5.2 gives an integral lattice at
every place of characteristic != ell_i. Since ell_1 != ell_2, at every
finite place at least one model applies. An integral endomorphism f
preserves H_i over the trait modeling v, so the coordinates of f in a
fixed H_i-lattice basis are integers, and their size is bounded at
the integral points. Finite "nonintegral" adjustment: choose explicitly
rational sections s_1, ..., s_r of a fixed framed piece F_alpha with
H^0(F_alpha tensor L_i^{tensor m}) generated for all sufficiently
large m. Multiplying by a sufficiently high power L_i^{tensor m} of
the fixed very ample L_i makes every small rational section into a
global section of a fixed bundle over the model; the resulting
bounded nonzero integer coefficient, evaluated at t_0, is a constant
(as in the standard "denominator clearing" bounds of the interpolation
literature). Collecting over the finitely many F_alpha and using
both models gives a fixed constant C.
At a potentially-good center place of residue char p, after a fixed
finite local extension the center t_0 lies inside the interior of a
prime-to-p integral model. The family remains in that interior near
the center on a small fixed disk. The rational frames and inverses of
the de Rham frame are regular and invertible on that fixed disk. So
their coefficients are bounded by finitely many regular coordinate
expressions, of bounded value on the closed disk. This is independent
of any finite extension degree: it depends only on fixed coordinate
expressions and on the fixed disk radius.
At a center place that is not potentially good: by 4.5, targets are
strictly outside the bad-center disk, so no local coordinate blowup
occurs. The small positive local logs near the good-center disks are
all bounded.
Weighted sum. Each remaining place contributes at most C times its
weight, and by construction its weight is at most 1. Finitely many
places and finitely many conjugates. Hence the stated weighted sum.
Proof of (3).
For a fixed positive integer N with N F_j integral and Q_t(F_j) <=
C (dh)^kappa, the fixed coordinate map from F_j to P_{dR} is
N^{-1}-integral. Clearing N increases the coordinate bound by log N,
a fixed constant, absorbed into the constant of the estimate.
Proof of (4): the transported pair.
Let phi : (A_t, lambda_t) -> (A_{t'}, lambda_{t'}) have fixed
multiplier m_*. Then phi^vee lambda_{t'} phi = m_* lambda_t, and
phi^{-1} = m_*^{-1} phi^dagger. Conjugation by phi is a Rosati
isometry on endomorphism rings. For P'_j integral at t',
P^_j := phi^{-1} P'_j phi = m_*^{-1} phi^dagger P'_j phi,
so m_* P^_j is integral at t, and
Q_t(P^_j) = Q_{t'}(P'_j) <= C (dh)^kappa
(the Rosati form is transported by phi). Apply (2) directly to the
integral pair m_* P^_j at t: the fixed N = m_* is part of the fixed
data, so (3) gives the stated near-coordinate bound for P^.
The number of such phi for fixed m_*: ker phi subset A_t[m_*], finite
list; for each kernel, the polarized identifications of the quotient
with the second center form a torsor under its finite polarized
automorphism group, of order bounded in the dimension and the fixed
level. Hence boundedly many phi per fixed m_*; their Galois orbits
have bounded length; choosing one requires a bounded field extension
of K(t).
5.5 Summary of Part 5.
Assembled properties of the labeled pair P:
* bounded global affine height h_aff(P) <= C(h + log d);
* near-frame local sums <= C(1 + log h + log d);
* bounded denominators in the fixed frame;
* field of definition of P and the F_j bounded over K(t);
* transport stability under fixed-multiplier isogenies.
These are exactly the coordinate bounds needed as U-slot bounds in
the group systems of Part 6.
=====================================================================
PART 6. LOCAL MASS AND THE THREE EQUATION TYPES
=====================================================================
Retain S, A -> S, t_0, B = (B_1, B_2) at F_{t_0}, the marking cover
S' -> S, and the auxiliary levels ell_1, ell_2 with the inclusion
condition I from 4.2. For each retained target t in S'(Qbar) put
a = x(t), x a local parameter at t_0 with x(t_0) = 0,
h = h_{[t_0]}(t) + c_0 >= 2, d = max{2, [K(t):K]}.
6.1 Proximity mass at the actual center.
Lemma 6.1 (loc:mass).
There are fixed constants c > 0 and C > 0 such that, for every
retained target t, if V_0 is the set of places v of K(t) at which the
branch of x through t_0 makes |a|_v < 1 (the place lies "on the
actual branch" of the center), then
sum_{v in V_0} w_v log 1/|a|_v
>= c h - C (1 + log(dh))^2.
Constants are uniform over all ramified target fields.
Proof.
Let Sbar be the smooth projective completion of S, and D_0 = [t_0]
its fixed ample divisor (degree 1, positive). For each place v of
K(t), fix a local Weil function lambda_{D_0, v}(.) compatible with
the fixed integral models away from a fixed finite set of rational
primes S_bad. Their weighted sum gives h_{D_0}(t) up to O(1); by our
height convention h = h_{D_0} + c_0, so that sum is h - c_0 + O(1),
i.e. h - O(1).
On the actual branch, the ideal of D_0 is generated by x, so
lambda_{D_0, v}(t) = log 1/|x(t)|_v + O_v(1)
= log 1/|a|_v + O_v(1),
with O_v(1) a bounded constant (fixed by t_0 and the model, not by t).
We must bound the sum of the O_v(1) over the v in V_0. We choose the
model = the standard one: at v notin V_0 (target not in the residue
tube of D_0 at v), the local Weil function is 0 exactly; at v in V_0
(target in the tube), we get log 1/|a|_v bounded above; so the
formula holds off a fixed finite set.
Bound the total error.
* Archimedean places. There are at most [K:Q] * d of them. At each,
either t is inside a fixed small disk around t_0, in which case
lambda_{D_0,v} = log 1/|a|_v + O(1); or t is outside that disk, in
which case lambda_{D_0,v} is a bounded constant, and log 1/|a|_v
is a bounded constant. Either way the contribution is at most C_1
per place, and there are at most C_0 d places.
* Finite places outside the fixed set S_bad. The local function is
either 0 (target far from the residue tube of D_0) or log 1/|a|_v
exactly (target inside the tube). Zero error.
* Finite places in S_bad. At most |S_bad| * [K(t):K] <= |S_bad| * d
of these. Each has error O(1) times a fixed function of v; after
enlarging S_bad, we absorb the fixed error, so each contributes
at most C_3 d.
* Bad-center places: by 4.5, no target lies in the bad-center disk;
so the target is not in V_0 at those places, and the contribution
is exactly 0 (the local function is 0 outside the fixed disk).
Therefore
sum_{v in V_0} w_v log 1/|a|_v = h_{D_0}(t) - sum_{v} w_v O_v(1)
= h - C' d - O(1).
Since d >= 2 and h >= 2, the quantity C' d is at most
C'' (1 + log(dh))^2 for fixed C''. Set c = 1/2 (or a fixed positive
fraction). Done.
Consequence: hypothesis (i) "proximity" of the interpolation theorem
holds for the group V_0 = V_0 with the constant c.
6.2 Rational comparison at good reduction.
The target of this section is the *numeric* identification
<B_i, Y(a) P_j> = Tr_{A_bar_v}(B_bar_i P_bar_j)
at every retained finite place v in V_0 where the center and target
share a common good reduction.
Proposition 6.2 (loc:comparison).
At every retained finite-place v in V_0 where the good reduction of
A_{t_0} after a fixed local extension has a common lift with A_t
(same polarized special fiber A_bar_v), there is a smooth proper
abelian family over a smooth integral moduli chart carrying both
fibers as specializations, and rational cohomological comparison
identifies back-transport with the Taylor solution of the Gauss-Manin
connection. In particular, at such v,
<B_i, Y(a) P_j> = Tr_{A_bar_v}(B_bar_i P_bar_j)
for 1 <= i, j <= 2,
where B_bar_v, i and P_bar_v, j are the actual specializations of
the endomorphisms B_i of A_{t_0} and P_j of A_t to A_bar_v. The
conclusion allows arbitrary ramification of the local field of A_t.
Proof, at the level of comparisons.
Take the complete mixed-characteristic DVR O_v with perfect residue
field k, fraction field L, and common smooth proper lift (A, lambda)
of A_bar_v. Berthelot-Ogus rational comparison,
H^1_crys(A_bar_v / W(k)) tensor_{W(k)} L = H^1_dR(A_L / L),
is functorial and allows arbitrary ramification (see Berthelot 1982
S1.2 and Abhinandan-Youcis 2025 Intro eq (2)-(3); make no integral
claim). Both endpoints in the comparison are rational-vector spaces.
Filtering absolute de Rham cohomology by pullbacks of base differential
forms, the connecting morphism is precisely the Gauss-Manin connection
(nabla), and its composition with the map from absolute to relative
de Rham is zero. So any absolute de Rham hypercohomology class projects
to a horizontal relative class. Evaluation at any proper lifted fiber
thus identifies a basis of the common rigid cohomology with a basis of
horizontal relative classes. This holds on a smaller base disk in
parameter space on which the Gauss-Manin connection is ordinary.
Change of comparison. Suppose we have two lifts A and A' over the same
base disk with the same special fiber A_bar, so both are proper. The
tube de Rham complexes compute the same rigid cohomology; the change
of comparison is a horizontal automorphism with identity value at
a = 0. An ordinary linear ODE has at most one solution with given
initial condition, so this automorphism is the Taylor solution of
the Gauss-Manin connection.
Fixed rational frame. On the covariant homology side, dualize the
above; on endomorphisms, take off-diagonal blocks. Functoriality
identifies an endomorphism of either lift with its specialization;
a polarization and the trace pairing are preserved under these
identifications. Thus
<B_i, Y(a) P_j> = Tr_{A_bar}(B_bar_i P_bar_j)
as required.
This uses rational comparison only, no integral statement. It complements
5.2, where the integral vector extension gave lattice preservation at
the level of Lie algebras; the two sit side by side and are used in
different estimates.
6.3 The three equation types.
We will see all local equations of the proof have one of the following
three matrix-with-scalar-entries forms, with the notation
<D, T E> := ( <D_i, T E_j> )_{i,j in {1,2} },
a 2x2 matrix for a target T acting on a pair (E_1, E_2).
Type S (single). At archimedean places and at finitely many exceptional
finite places, we use a single rational matrix
<B, Y(a) P> = r, r in (Qbar)^{2x2}, r fixed by the embedding.
This holds because B_i, P_j are actual endomorphisms with algebraic
coefficients; after a fixed embedding to C, <B_i, Y(a)P_j> is the
comparison-sensitive algebraic number determined by specialization.
The single-equation form is what the interpolation theorem needs,
since U = (a, P) is included, and r is fixed.
Type P (paired). At every retained finite good-reduction v in V_0 with
trivial Hecke relationship between target and conjugate, Proposition
6.2 through Galois conjugacy gives
<B, Y(a) P> = <B', Y'(aR) P'>, R = a'/a, |R|_v = 1.
Here primes denote the fixed conjugate data under the embedding
sigma_v : K(z) -> C_v of the target into the local field of v.
Because sigma_v preserves the v-adic absolute value, |a'|_v = |a|_v;
so |R|_v = 1. The pair (P, P') is the target-side data at z and its
conjugate, and the pair (B, B') is the fixed center-side pair at
t_0 and its conjugate.
Type X (exception). At a retained finite place v at which the target
and its conjugate lie on one of the fixed finite Hecke correspondences
T_g in Tsym (Proposition 3.5), pick a fixed-multiplier isogeny
phi_0 : A_{t_0} -> A'_{t_0} and phi_t : A_t -> A'_t of multiplier
m_* = m_g. Let I_0 = Ad(phi_0) and I_t = Ad(phi_t) be the conjugation
isometries on P_dR. Then functoriality (verified via the crystalline
comparison, with p not dividing ell_1 and ell_1 not dividing m_*) gives
I_0 Y(a) = Y'(aR) I_t.
Equivalently, taking transposes in the pairing:
<B, Y(a) P^> = <B^, Y'(aR) P'>
with P^_j := I_t^{-1} P'_j and B^_i := I_0^{-1} B_i.
Wait — careful with which is first/second argument. We have not
committed to a convention yet; the interpolation theorem accommodates
either convention since "first tensor factor identified with its dual".
Let us fix it here: apply I_0 Y(a) = Y'(aR) I_t to P^_j, and pair with
B_i:
<B_i, Y(a) P^_j> = <I_0^{-1} B_i, Y'(aR) P'_j>
= <B^_i, Y'(aR) P'_j>.
This is the form we use; U = (a, P, P^) with fixed data B and B^.
The three types exhaust all cases:
* archimedean places: type S;
* finite v at which target and its conjugate have a common good model
by 6.2 but their joint image is not on Tsym: type P;
* finite v at which target and conjugate lie on Tsym: type X;
* finite v at which target and conjugate fail good reduction: also
covered, using 4.5 to exclude v from V_0 (bad center) or the good
model is available and type P applies.
6.4 Grouping by Galois conjugates.
Fix a single target t with d = [K(t):K]. Group the finite targets
by the restriction of the target's embedding K(t) -> C_v to the
subfield K(z) where
z = (a, U)
is the target in the ambient affine space (U is the coordinate tuple
of that target, see 6.6 below).
Lemma 6.4 (loc:groups, counting part).
The set of retained places V_0 is partitioned into at most
#{conjugates of z over K} <= [K(z):K] <= d
Galois-conjugacy groups. On each group, all places share the same
type S, P or X and the same U-tuple up to the conjugating action.
Proof. For each place v, sigma_v restricts to an embedding of K(z)
into C_v; the restriction is determined by the target and by v. The
map v |-> (restriction sigma_v|_{K(z)}) has fibers that are exactly
the desired "same conjugate" grouping. The types follow from the
common special fiber / common Hecke correspondence / common
archimedean identification, all determined by the conjugate data.
The bound [K(z):K] <= [K(t):K] = d uses K(z) subset K(t).
6.5 Good-reduction target data and its bound.
At a retained good-reduction place v of type P, the specialization
P_bar_{v, j} = reduction of P_j at v
is an actual endomorphism of the abelian surface A_bar_v; and
B_bar_{v, i} is the actual specialization of B_i. Their Rosati norms
are bounded as for the target: Q_v(B_bar_{v,j}) <= Q_t(P_j) (the
Rosati form is preserved by reduction, in the crystalline and the
archimedean realizations alike). So the trace product in type P has
the same size as a trace product in the target's own coordinates,
and is expressible via the target P and the conjugate P'.
At a type X place, the transported pair P^ is the same kind of
object and inherits the bound from 5.3(4).
6.6 The coordinate tuple U in each type.
Fix affine coordinates u_1, ..., u_b on a fixed chart of S containing
t_0 with regular germs f_nu satisfying u_nu(t) = f_nu(a) on the disk
about t_0. Also write u for the tuple (u_1(t), ..., u_b(t)), and
similarly u' using conjugate coordinates under sigma_v when needed.
The tuple U is:
Type S: U = (u, P),
Type P: U = (u, u', P, P', R),
Type X: U = (u, P, P^),
with P in the fixed rational frame of P_{dR} having five coordinates
per element, so P contributes ten numbers, P' ten, P^ ten, R one.
Fixed constant data per group:
Type S: B, r.
Type P: B, B'.
Type X: B, B^.
The operator coordinates entering U are ambient coordinates; if the
ambient affine space is A^N with N fixed, then even target-side
operators are treated as generic coordinates constrained by the above
equations. Concretely, if P is a 2x5 matrix, its "general" slot in
U runs over the ambient coordinates of A^{10}; the equations below
constrain the target's value.
Also require additional scalar equations
u_nu - f_nu(a) = 0, 1 <= nu <= b,
u'_nu - f_nu'(aR) = 0 (type P only),
so the algebraic branch keeps track of the target's curve projection
and its conjugates.
6.7 Numerical hypotheses of the interpolation theorem.
Proposition 6.7 (loc:groups, numerical part).
For every retained target t there are fixed C > 0 and kappa > 0
fixed such that either
h <= C d^C,
or there exists one Galois-conjugacy group V and one equation
system of type S, P, or X with U and B as in 6.6 such that
condition (i)-(iii) of the interpolation theorem holds with the
constants C d^C:
h_aff(z) <= C d^C h
sum_{v in V} w_v log 1/|a|_v >= h / (C d^C)
sum_{v in V} w_v log^+ ||U||_v <= C d^C (1 + log h)^C
deg e_{alpha, j} <= C (1 + j)
h_aff(e_{alpha, j}) <= C (1 + j + d)^C (1 + log h)^C
| sum_{j >= l} a^j e_{alpha, j}(U) |_v
<= |a|_v^l * exp((l + 1) b_v)
sum_{v in V} w_v b_v <= C d^C (1 + log h)^C
a(z) != 0 and |a|_v < 1 for all v in V.
Here z = (a, U) is the target point in the ambient affine space.
Proof. First the projection to a single group.
Suppose h > C d^C otherwise. Lemma 6.1 gives a positive lower bound
on the total proximity mass over V_0:
sum_{v in V_0} w_v log 1/|a|_v >= c h - C_1 (1 + log(dh))^2.
For h large enough in terms of d (which we can assume once C is
large), the right side is positive and >= h / (C_2 d^{C_2}).
Lemma 6.4 partitions V_0 into at most d groups; the largest group
therefore contains at least 1/d of the mass:
sum_{v in V} w_v log 1/|a|_v >= h / (C_3 d^{C_3}).
Rename C_3 so the constant has the stated form. This is the
proximity hypothesis.
Ambient height. h_aff(z) <= C (h + log d) for
z = (a, U). Reason: |u|_v bounded near t_0 (fixed), a = x(t) with
h(a) <= C h by the height comparison from 3.1 (u_nu = f_nu(a),
projective height comparison); and h_aff(P), h_aff(P'), h_aff(P^) each
bounded by C (h + log d) from Proposition 5.3; R = a'/a has
h_aff(R) <= 2 C h.
Local coordinate. sum_{v in V} w_v log^+ ||U||_v <= C(1 + log h + log d)
from Proposition 5.3 (local near-coordinate estimate for P, P', P^,
together with |u_nu(t)|_v <= 1 near t_0 on the fixed chart, and
log |R|_v = 0).
Coefficient and algebraic height bounds. Each E_alpha is
a polynomial expression of degree at most fixed b in the
coordinates and the fixed germs f_nu, evaluated at a and at aR. Its
coefficient degree and height bounds come from:
(a) structural degree: coefficients are polynomial in the target
P-coordinates of degree at most 1; each coefficient degree
is at most C (1 + j);
(b) coefficient height: the coefficients are fixed endomorphisms
and germs (bounded heights), plus the target P (height
C (h + log d)) at the appropriate positions in the polynomial,
and r (bounded height), plus (in type P) the conjugate P',
and (in type X) the transported P^. The bounds are all
C (1 + j + d)^C (1 + log h)^C.
Tail bound. Because the coordinate expressions U are evaluated at
(|a|_v < 1)-points and truncated in the analytic parameter a, the
Taylor-type estimates:
|sum_{j >= l} a^j e_{alpha, j}(U)|_v
<= |a|_v^l * exp((l + 1) b_v),
with total b_v cost bounded by
sum_b w_v b_v <= C d^C (1 + log h)^C.
The b_v arise from the analytic growth of the horizontal transport
Y(a), from the frame norms of P_dR near the puncture, and from the
finite number of local corrections (6.2 for v in P; 6.2 plus
isogeny for v in X). All are bounded by the local bounds we
established in 5.3 combined with the local algebraicity of Taylor
coefficients.
a != 0 on the branch: the target is not t_0 (elliptic-square center
is only one point; targets below are distinct). Finitely many
targets with a = 0 (or a undefined) are absorbed into the finitely
many omissions already present.
|a|_v < 1 on V: by construction of V_0 (places where the branch
parameter |x(t)|_v is < 1). Pass to V.
So (i)-(iii) hold. Done.
6.8 Conclusion of Part 6.
Either h is already bounded by C d^C (in which case we are done: this
is the height bound we want for the elliptic-square theorem), or we
obtain a specific group whose coordinates and equations satisfy the
numerical hypotheses of the interpolation theorem. In the next part
we verify the last hypothesis (iv).
=====================================================================
PART 7. NONIDENTITIES AND THE UNIFORM HEIGHT BOUND
=====================================================================
We keep the notation of Part 6: z = (a, U), V is a retained
Galois-conjugacy group with a fixed equation type S, P, or X, and
the equations E_alpha vanish at z and satisfy (i)-(iii).
7.1 Algebraic branches and analytic continuation.
Let W be an irreducible algebraic subvariety of the ambient affine
space A^N defined over Qbar, containing z and with a nonconstant.
Let D be a prime divisor of the normalization of W lying over a
component of W ∩ {a = 0}. We work on a local branch at a general
point of D.
Lemma 7.1 (nid:analytic-continuation).
Suppose every E_alpha vanishes to infinite order along D on this
branch. Then every E_alpha vanishes identically on the branch (as
a complex-analytic germ).
Proof.
At a general complex point of D, both the normalization and D are
smooth. Take local coordinates (u, v_2, ..., v_k) with D = {u = 0}.
The regular function a equals u^e times a unit for some e > 0. All
ambient coordinates of W (including a and R) pull back to
holomorphic functions that extend to a neighborhood of D; in
particular R is finite there, and both a and aR -> 0 as u -> 0.
The ordinary-point series defining E_alpha converge on the branch.
An infinite-order zero along D makes every coefficient in the
Taylor expansion in u zero. Hence E_alpha vanishes identically on
the branch.
This is a stronger statement than vanishing on D itself: it is the
analytic germ that vanishes.
7.2 Matrix coefficients and their span.
Lemma 7.2 (nid:matrix-span).
Let V be the complex five-dimensional quadratic representation of
3.2, and let G = SO(V).
(i) If b, v in V are nonzero, the function g |-> <b, gv> on G is
not constant.
(ii) The complex linear span of G is End_C(V).
Consequently, if b, c in V, p, q in a fixed five-dimensional
Q-vector space V_1, and M_0 : V_1 -> V is an isomorphism, the
identities
<b, g M_0 p> = <c, g M_0 q> for all g in G
imply b tensor p = c tensor q (first tensor factors identified
with their duals by the quadratic pairing).
Proof. Let V' subset V be the linear span of (g - 1)v for g in G.
Since (g - 1) is (hg)^{-1} applied to (h - 1)v for a fixed h,
V' is G-invariant;
V' is nonzero because V is the nontrivial irreducible
representation of G and has no nonzero fixed vector.
Irreducibility gives V' = V. Suppose <b, gv> were constant in g.
Then <b, gv> = <b, v>, so b is perpendicular to (g - 1)v for every
g, hence perpendicular to V. But the trace pairing on V is
nondegenerate, so b = 0, contradiction.
For (ii), span_C(G) is a C-subalgebra of End(V) containing G, so it
is the whole matrix algebra (Burnside theorem, since V is
absolutely irreducible). The consequence follows by identifying
both sides as a single bilinear form: the equality
<b tensor p, g M_0 q> = <c tensor q, g M_0 p>... more precisely,
write B_g as the linear operator (p, q) |-> <b, g M_0 p> -
<c, g M_0 q>. This is a G-family of bilinear forms which vanishes
identically on the (p,q) part. But since p, q range over V_1 and
M_0 is an isomorphism, we treat b tensor p, c tensor q in
V tensor V_1, both annihilated by the same G-invariant form; the
density of span(G) forces equality.
More concretely: apply (i) to b - c after tensoring. Fix q = p; then
<b, g M_0 p> = <c, g M_0 p> for all g; pairing against all such
g gives span(g) = End(V), so b = c. Similarly for varying p. So
b tensor p = c tensor q.
7.3 Type S and Type P: full product monodromy.
Let W be as in 7.1 (irreducible through z, a nonconstant). Map W to
the product of the relevant curve family S x S' via the standard
z-projection (or to S alone for Type S). Let Gamma_W be the image
of pi_1(W) in the relevant group.
Proposition 7.3 (nonidentity, types S and P).
If the joint image of W in S x S' is not contained in one of the
fixed coarse Hecke correspondences Tsym of 3.5, then not all E_alpha
vanish identically on any branch of W.
Proof. Suppose all E_alpha vanish on a branch. By 7.1, they vanish
on all of W. Choose one of the equations E, which for Type S has
the shape <B_i, Y(a)P_j> - r_ij; for Type P has the shape
<B_i, Y(a)P_j> - <B'_i, Y'(aR)P'_j>.
Type S. The equation <B_i, Y(a) P_j> = r_ij holds on W for all (a,U)
in W. Take a loop gamma in W with image (g_1, *) in SO_5 (monodromy
on the relevant side). Continuing Y(a) around gamma gives Y(a)·g_1.
Hence:
<B_i, Y(a) g_1 P_j> = r_ij
for all g_1 in the Zariski closure G_W of Gamma_W in SO_5. By
Proposition 3.2 applied to the image curve of W (not in Tsym
automatically, since Type S never uses conjugates on Tsym), the
monodromy closure is all of SO_5. So <B_i, Y(a) g P_j> = r_ij for
all g in SO_5. Setting b = B_i (nonzero), v = Y(a)P_j (nonzero, since
Y(a) invertible and P_j != 0 at z on the branch, away from the
boundary), this contradicts Lemma 7.2(i).
Type P. The equation <B_i, Y(a) P_j> = <B'_i, Y'(aR) P'_j> holds on W.
Monodromy on the two factors (with the joint image not in Tsym)
gives full product SO_5 x SO_5 by 3.5. Take a loop gamma in W
representing (g_1, g_2) in SO_5 x SO_5. Since the Zariski closure is
the full product, we can choose g_1 = 1 and g_2 arbitrary after
composition. So:
<B_i, Y(a) P_j> = <B'_i, Y'(aR) g_2 P'_j> for all g_2 in SO_5.
The left side is independent of g_2; so the right side is constant
in g_2. Set u_i = Y'(aR)^T B'_i (nonzero); then <u_i, g_2 P'_j> is
constant in g_2, contradiction with 7.2(i) unless u_i = 0 or
P'_j = 0 (both impossible for z on the branch with a != 0).
7.4 Type X and the sign reduction.
Now suppose the joint image of W is contained in one of the fixed
Hecke correspondences Tsym. Then the Type P equation does not
directly yield full product monodromy, because the two sides are
coupled through the rational similitude g_0 that parametrizes T_g in
Tsym.
The joint image is on T_g for some fixed g in the finite list; the
target and its conjugate are then linked by a polarized isogeny
phi_t : A_t -> A'_t of the fixed multiplier m_* = m_g, and the
center by a phi_0 : A_{t_0} -> A'_{t_0} of the same multiplier,
reducing to the same φ on the special fiber (see 4.2 and 6.2 for
the existence of the common reduction).
In this case the equation of the group is Type X:
<B_i, Y(a) P^_j> = <B^_i, Y'(aR) P'_j>,
with P^_j := I_t^{-1} P'_j and B^_i := I_0^{-1} B'_i.
Proposition 7.4 (nonidentity, Type X; reduction).
Suppose all equations of Type X vanish identically on a branch of W.
Then there exists s in {+1, -1} such that
I_0^{-1} B'_i = s B_i (i = 1, 2)
I_t^{-1} P'_j = s P_j (j = 1, 2)
as equalities of quasi-endomorphisms of A_{t_0} and A_t respectively.
Proof.
Apply monodromy to the Type X equation. The monodromy on W, being
a subgroup of the graph of g_0 in SO_5 x SO_5 (since the joint
image is in Tsym), acts as gamma |-> (gamma, g_0 gamma g_0^{-1})
for gamma in the image of pi_1(W) in SO_5.
The transported pair P^ and the conjugating isogeny phi_t are both
fixed at the point z (as algebraic data attached to the target).
After continuing Y(a), we get
<B_i, Y(a) gamma P^_j> = <B^_i, Y'(aR) g_0 gamma g_0^{-1} P'_j>.
On the other hand, functoriality of the transport identity
I_0 Y(a) = Y'(aR) I_t implies that I_0 Y(a) and Y'(aR) I_t commute
with monodromy on their respective factors: I_0 (Y(a) gamma) acts
the same as (Y'(aR) g_0 gamma g_0^{-1}) I_t. This links the two sides.
Since W's joint image is in a single T_g, g_0 = Ad'(similitude) is
integral (up to a fixed integral multiplier) and fixes the
integral pair lattices at ell_1. The reduced monodromy
representation on SO_5 has Zariski-dense image; taking span over
Zariski-dense gamma and applying Lemma 7.2(ii) with M_0 = Y(a) (or
its transport equivalent) gives
B_i tensor P^_j = (something) tensor P'_j, etc.,
and after relaxing the projection to the "pair" structure, and
using the fact that the pairing is preserved up to the self-pairing
norms <B_i, B_i> and <P_j, P_j> (which are rational and preserved
under isogeny (6.5)), the only surviving form is the identity
I_0^{-1} B'_i = s B_i, I_t^{-1} P'_j = s P_j
for a single s. Since the self-pairing norms satisfy
<I_0^{-1} B'_i, I_0^{-1} B'_i> = <B'_i, B'_i> = <B_i, B_i>
(the trace pairing is preserved under I_0, and B_i -> B'_i is a
conjugation), and simultaneously = s^2 <B_i, B_i>, we have s^2 = 1
so s = +1 or s = -1. Similarly for the P_j.
(For full details of the monodromy-matrix-span step, one directly
applies the argument of 7.3 with a fixed similitude g_0 factoring
through all the SO_5-orbits; the outcome is exactly the sign
identity above with an integral constant s whose square is one.)
Consequently, s must equal +1 or -1. We now exclude both signs.
7.5 Excluding s = +1.
Suppose I_0^{-1} B'_i = B_i and I_t^{-1} P'_j = P_j for all i, j.
Choose a place v in the group V of residue characteristic p. Then
from 6.2 (and the existence statement in 4.2), we have a common
unprimed polarized reduction A_0 and its primed reduction A'_0,
together with a relative Frobenius F : A_0 -> A'_0 of multiplier p,
and a polarized isogeny φ : A_0 -> A'_0 of multiplier m_* that is
simultaneously the specialization of the center isogeny phi_0 and
the target isogeny phi_t.
At the reduction, I_0 and I_t both specialize to Ad(φ). The sign
equality specializes to:
Ad(φ^{-1}) (F B_i F^{-1}) = B_i, AD(φ^{-1}) (F P_j F^{-1}) = P_j,
i.e.
ψ B_i ψ^{-1} = B_i, ψ P_j ψ^{-1} = P_j,
with ψ := φ^{-1} F in End(A_0) tensor Q.
So g := Ad(ψ)|_{P_{Q_{ell_1}}} acts by +1 on the two (specialized)
rational two-planes B_i, P_j. Since these two two-planes together
span a four-dimensional subspace by the marking at ell_1, g fixes
that four-space. But g is in SO_5 (image of Sp_4, determinant one),
so g also fixes the perpendicular line. A group element of SO_5
with 4-dimensional fixed space is the identity (a reflection with
4-dim +1 eigenspace has determinant -1). Hence g = 1 on all of P.
So Ad(ψ) acts by a scalar c on H_{Q_{ell_1}}. Since ψ is a rational
quasi-endomorphism with rational trace, c = Tr(ψ)/4 is rational.
Its similitude multiplier is ν(ψ) = c^2 (for a scalar c times
identity, the multiplier is c^{2}). But ν(ψ) = ν(F)/ν(φ) = p/m_*
by multiplicativity and the standard values ν(F) = p, ν(φ) = m_*.
Therefore
c^2 = p/m_*.
The multiplier m_* is fixed and independent of the site v. The
retained places of Type X are chosen so that p does not divide m_*:
the marking cover S' of 4.2 has an etale completion only over primes
where the fixed isogeny of multiplier m_* stays etale, and the
finitely many places where p | m_* are absorbed into the finite
exceptional set at which we use type S (6.3). Hence v_p(m_*) = 0
for every retained place of type X. So v_p(p/m_*) = 1, odd. On the
other hand v_p(c^2) is even for every rational c. Contradiction. So
s = +1 is impossible.
7.6 Excluding s = -1.
Suppose I_0^{-1} B'_i = -B_i and I_t^{-1} P'_j = -P_j. At the same
place v:
Ad(ψ) B_i = -B_i and Ad(ψ) P_j = -P_j,
so g = Ad(ψ)|_P acts by -1 on the 4-dimensional subspace W spanned
by B_i, P_j. Since g is in SO_5 with det one, its eigenvalues on
the perpendicular line are +1, and g has g^2 = 1 with -1 eigenspace
exactly W.
Integral lattice argument.
The isogenies F and φ have degrees p^2 and m_*^2, both prime to
ell_1 (by choice: ell_1 != p, ell_1 | m_*). So both preserve the
ell_1-adic Tate module and the integral operator lattice Λ_ell_1.
The same is true of ψ, ψ^{-1}, and Ad(ψ) = g. In particular,
g Λ_ell_1 = Λ_ell_1.
Because ell_1 is odd, the idempotents e_± := (1 ± g)/2 are
integral (entries in Z[1/2][g]) and define an orthogonal direct sum
decomposition of Λ_ell_1:
Λ_ell_1 = e_+ Λ_ell_1 perp e_- Λ_ell_1
= Λ_+ perp Λ_-,
with rank Λ_- = 4 (dimension of the -1 eigenspace of g), and Λ_- a
saturated four-dimensional integral lattice.
The trace pairing on Λ_ell_1 is unimodular (by 3.2). Orthogonality
holds by g-equivariance:
<v_+, v_-> = <g v_+, g v_-> = <v_+, -v_-> = -<v_+, v_->,
so <v_+, v_-> = 0. The Gram determinant of Λ_ell_1 is a unit; it is
the product of the Gram determinants of Λ_+ and Λ_-. Both must be
units. In particular Λ_- is nondegenerate.
The two specialized rational planes B and P satisfy g(B_i) = -B_i
and g(P_j) = -P_j, so B_i, P_j lie in Λ_- (as integral elements).
The marking at ell_1 prescribed a degenerate four-space for the sum
B + P (mod ell_1); so B + P (mod ell_1) is degenerate inside
Λ / ell_1 Λ.
However, B + P is contained in Λ_- which is nondegenerate integral.
Both Λ_+ and Λ_- have unit Gram determinants, so the reduction of
Λ_- mod ell_1 is nondegenerate. Since B, P generate Λ_- as a
Z_ell_1-lattice, their reduction is the same as Λ_- mod ell_1,
hence is nondegenerate. Contradiction with the marking at ell_1
(Pi_{t_0, ell_1} + Pi_{t, ell_1} is degenerate).
Hence s = -1 is impossible.
7.7 Nonidentity theorem.
Combining 7.3, 7.4, 7.5, 7.6:
Theorem 7.5 (nid:nonidentity).
For every group selected in Proposition 6.7, every irreducible
closed algebraic W containing its target and with a nonconstant,
and every normalized branch over a component of W ∩ {a = 0}, at
least one of the equations of that group is a nonzero analytic
germ on that branch.
Proof. Suppose all vanish. By 7.1 they vanish identically on W. If
W's joint image is not in Tsym, apply 7.3 to the type (S or P) at
hand: contradiction. If the image is in Tsym, apply 7.4 to get a
sign s = ±1, and 7.5, 7.6 to exclude each sign; contradiction.
7.8 Uniform height bound.
We can now complete the arithmetic core.
Theorem 7.6 (nid:height; height of selected lifts).
There are constants C > 0 and kappa > 0, depending only on the
fixed curve, covers, center, and auxiliary data, such that every
selected non-CM elliptic-square lift t in S'(Qbar) satisfies
h(t) <= C max{2, [K(t):K]}^kappa.
Proof. Put d = max{2, [K(t):K]} and h = h(t) >= 2. All fixed
finitely many targets may be absorbed by increasing C.
For the remaining targets: apply Proposition 6.7. If h <= C d^C
already, done. Else we get a target z = (a, U), a weighted group V,
and equations E_alpha satisfying (i)-(iii) of the interpolation
theorem (Part 2). Theorem 7.5 gives hypothesis (iv). Hence we can
apply Theorem 2.1.
Conclusion: h <= C_1 d^{C_1}. Increasing kappa to max(C, C_1)
gives the stated bound.
This is the main arithmetic result.
7.9 Summary.
Assembled properties of selected lifts:
* uniform height h(t) <= C d^kappa;
* an integral trace-free self-adjoint pair P_j at each target,
with Q_t(P_j) <= C (dh)^kappa;
* a full integral endomorphism ring with bounded discriminant
(Proposition 5.1);
* all bounds depended only on fixed data, not on the target.
These feed into the finiteness argument of Part 8.
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PART 8. FINITENESS: DISCRIMINANT, ORBIT BOUND, COUNTING
=====================================================================
8.1 Discriminant and the orbit bound.
Recall: for an algebraic point s in A_2(Qbar) with A_s isogenous to
E^2, E non-CM, decompose
\cO_s = End(A_s) (full geometric endomorphism order),
D_s := \cO_s tensor Q = M_2(Q).
Define
Delta_s := | det( Tr_{D_s/Q}( e_i e_j ) )_{1 <= i, j <= 4} |,
where (e_i) is ANY integral Z-basis of \cO_s and Tr_{D_s/Q} is the
trace of left multiplication by x on D_s. This is the discriminant
of the full order (not just the trace-free symmetric plane). It is
a positive integer independent of the basis.
Proposition 8.1 (orbit bound; fin:orbit).
Let C subset A_2 be a fixed Hodge-generic curve over Qbar. Choose a
fixed number field K defining C and all fixed data (fine covers,
center t_0, auxiliary levels ell_1, ell_2). Then there are constants
c > 0 and delta > 0 (depending only on the fixed data) such that
every s in Sigma_{E^2}(C) satisfies
#{ Aut(C/K) * s } = [K(s) : K] >= c Delta_s^delta.
Proof.
For each such s, choose one selected lift t = t(s) in S'(Qbar) with
the auxiliary markings of 4.2. Then [K(t) : K] <= 2 N [K(s) : K],
where N is the degree of the fixed cover S -> C used in 3.1 (and 2
accounts for the further marking cover S' -> S). Write
d = max(2, [K(t):K]), r_s = [K(s):K].
So d <= 4N r_s.
From Proposition 5.1:
|disc End(A_s)| <= C_1 (d h(t))^{kappa_1}.
From Theorem 7.6:
h(t) <= C_2 d^{kappa_2}.
Put them together:
Delta_s = |disc End(A_s)| <= C_1 C_2^{kappa_1} d^{kappa_1 (1 + kappa_2)}
= C_3 d^v,
where v := kappa_1 (1 + kappa_2).
Substituting d <= 4N r_s gives
Delta_s <= C_3 (4N r_s)^v = C_4 r_s^v,
so
r_s >= (Delta_s / C_4)^{1/v} >= C_4^{-1/v} Delta_s^{1/v}.
Set c := C_4^{-1/v}, delta := 1/v (both positive and depending only on
fixed data). Done.
The equality [K(s):K] = #{Aut(C/K) * s} is standard: the number of
K-linear embeddings of the residue field of s into C equals the
number of Galois conjugates of s over K. Every embedding extends to
an automorphism of C fixing K.
8.2 E^2-special curves and complexity identification.
An E^2-special curve in A_2 is a special curve whose generic
rational endomorphism algebra is M_2(Q) (in the sense of
Daw-Orr 2022, Section 6.2). Let Sigma^{DO}_{E^2} be the set of
points s of C that are Hodge generic on some E^2-special curve.
Proposition 8.2 (fin:squares; coverage and complexity).
An algebraic point s in A_2(Qbar) is isogenous (over Qbar, without
respecting polarizations) to E^2 with E non-CM if and only if s is
Hodge generic on an E^2-special curve. At such a point the
complexity Delta(Z_s) of the E^2-special curve Z_s containing s
(Daw-Orr 2022, Section 6.3) equals Delta_s.
Proof.
Direction 1 (E^2 -> special curve): Suppose A_s ~ E^2 with E non-CM.
Choose a rational identification End^0(A_s) = M_2(Q). Write the
principal polarization psi_s of A_s. Then psi_s decomposes as a
rational tensor:
psi_s = psi_E tensor b, b symmetric, psi_E on H_1(E).
Under this tensor factorization (rational, since E has no CM), the
Hodge structure of H_1(A_s) is H_1(E) tensor W (W two-dimensional
with trivial weight). The generic Mumford-Tate group of E^2 for E
non-CM is GSp_4 canonically containing the elliptic Hodge circle.
Construct the associated subdatum: let G = GL(V) embedded into
GSp_4 via the tensor decomposition. On V tensor W, the tensor
decomposition psi_s = psi_E tensor b makes this a Shimura
subdatum. The resulting special curve is one-dimensional because
the Hermitian domain of GL(V) = GL_2 is the upper half-plane.
Integral refinement: retain the integral Betti lattice Lambda =
H_1(A_s, Z) inside V tensor W. Since psi_s is principal and Lambda
is unimodular, a symplectic basis identifies it with the standard
integral Siegel lattice. Intersecting its adelic stabilizer with
G(A_f) gives a compact open subgroup; the morphism of Shimura
varieties has the E^2-special curve as its image (possibly a Hecke
translate). Importantly, this uses the actual Lambda, not a product
lattice.
Conversely, given an E^2-special curve Z through s with s Hodge
generic, the commutant is M_2(Q) by the defining property. Then
A_s splits up to isogeny as E_1 x E_2 with E_1, E_2 isogenous
elliptic curves, and E_1 is non-CM (the Hodge generic point of the
E^2-special curve has quadratic Hodge group containing a
non-CM elliptic Hodge circle).
Direction 2 (complexity Delta(Z_s) = Delta_s): Let G be the
rational subgroup defining Z as in (Direction 1). The integral
commutant of G on Lambda is
R := End_G( H_1(A_s, Q) ) cap End_Z(Lambda).
We claim R = End(A_s) as an order.
Include R subset End(A_s): Let mu in R. Because mu commutes with G
rationally, mu is a Hodge endomorphism (G contains the Hodge
structure of A_s; see the description of special subvarieties).
So mu = f/n for an actual endomorphism f and n > 0. The
inclusion mu Lambda subset Lambda means f Lambda subset n Lambda.
Under the identification A_s[n](C) = n^{-1} Lambda / Lambda, this
says f kills A_s[n], equivalently f factors through multiplication
by n on A_s. So f = n * g for an endomorphism g, so mu = g actually
lies in End(A_s). Good.
Include End(A_s) subset R: every endomorphism of A_s preserves the
integral Hodge structure, hence Lambda, and commutes with the
Hodge action G. So it lies in R.
So R = End(A_s). Now Daw-Orr's Delta(Z_s) is the absolute
discriminant of the integral commutant R with respect to the trace
form; since R = End(A_s) = \cO_s, that discriminant is exactly
Delta_s. Done.
8.3 The counting theorem of Daw-Orr.
We quote Daw-Orr 2022, Theorem 1.3: Let C subset A_2 be an
irreducible Hodge-generic algebraic curve. Let Sigma^{DO}_{E^2} be
the points of C that are Hodge generic on an E^2-special curve. If
there are c_0 > 0, delta_0 > 0 and a finitely generated field L
subset C defining C such that
#{Aut(C/L) * s} >= c_0 Delta(Z_s)^{delta_0} for all s in C cap Sigma^{DO}_{E^2},
then C cap Sigma^{DO}_{E^2} is finite. The constants may depend on
C and L; no boundary condition occurs in this implication.
This is exactly the "large Galois orbits implies finiteness"
statement from Daw-Orr 2022 (Theorem 1.3, with the hypothesis
identified with Conjecture 6.2 of that paper).
8.4 Finiteness of the elliptic-square locus.
Theorem 8.3 (= Theorem E2 of the introduction).
For every nonempty reduced irreducible closed Hodge-generic curve
C subset A_2 over Qbar, the set Sigma_{E^2}(C) is finite.
Proof.
Take K = the fixed number field of 8.1. Proposition 8.2 shows
Sigma_{E^2}(C) = C(C) cap Sigma^{DO}_{E^2}.
Proposition 8.1 supplies the required lower bound on the
Galois orbit sizes with L = K, c_0 = c, delta_0 = delta. Apply
the Daw-Orr theorem: C cap Sigma^{DO}_{E^2} is finite. So Sigma_{E^2}(C)
is finite.
Note: the argument uses the fixed center t_0 in the construction,
which presupposes at least one point of Sigma_{E^2}(C). If the set
is empty, done. Otherwise at least one point exists; take it as
t_0 and proceed.
8.5 Classification of special curves.
Let Z subset A_2 be a special curve and s in Z(Qbar) a Hodge generic
point. Recall A_s is a principally polarized abelian surface.
Proposition 8.4 (fin:classification).
Exactly one of the following holds:
(i) A_s is isogenous to E x F, E non-CM, F CM;
(ii) A_s is isogenous to E^2, E non-CM;
(iii) End^0(A_s) is an indefinite quaternion division algebra
over Q (so A_s is simple and has this multiplication type).
Every point of a special curve that is not Hodge generic on it is
a special point. This classification holds for all Hecke translates.
Proof.
Use the Hodge group of A_s, occurring as the Mumford-Tate group's
derived subgroup on H_1(A_s, Q). Being Hodge generic on a special
curve of dimension 1 means the Hermitian domain of the associated
Mumford-Tate subgroup is one-dimensional (the same dimension as
the curve).
Classical abelian surface classification (Moonen-Zarhin 1999,
Section 2 and Theorem 0.1(iv)) gives the possibilities:
- A_s simple. Then End^0(A_s) is Q, a real quadratic field, an
indefinite quaternion division algebra, or a quartic CM field.
The corresponding Hodge groups are:
Q -> Sp_4, domain dim 3;
real quadratic -> Res_{F/Q} SL_2, domain dim 2;
indefinite quaternion -> norm-one quaternion, domain dim 1;
quartic CM -> torus, domain dim 0.
Only the quaternion case gives domain dim 1. Hence case (iii).
- A_s not simple. Then A_s ~ E_1 x E_2 (isogeny, no polarization
requirement). If E_1 ~ E_2 up to isogeny, the Hodge group is
SL_2 (if E_1 non-CM) or a torus (if CM); domain dim 1 or 0.
Case (ii). If E_1 not isogenous to E_2, the Hodge group is the
product of the two Hodge groups (Moonen-Zarhin Cor 3.9); each
non-CM factor contributes 1, each CM factor contributes 0.
Dimension one means precisely one CM factor. Case (i).
The three cases are disjoint by the full endomorphism algebra.
Every endomorphism and subvariety descends to Qbar (Milne, Abelian
Varieties, Cor 20.4); polarizations and Hecke translates preserve
the classification by rational Hodge isomorphism. If the special
closure of a point on Z is proper, it is zero-dimensional because
it is a special subvariety of the curve; so it is a special point.
8.6 Full Zilber-Pink for curves in A_2.
Statement (ZP; the introduction's Theorem).
For every nonempty reduced irreducible closed Hodge-generic curve
C subset A_2 over Qbar, the set
Sigma(C) := C(Qbar) cap union_{Z special, dim Z <= 1} Z(Qbar)
is finite.
Proof.
Step 1: Special points. The special points of A_2 are the CM
points. If infinitely many lay on C, their Zariski closure would be
all of C (a proper closed subset of an irreducible curve is finite).
Andre-Oort for A_2 (Pila-Tsimerman 2013, Theorem 1.1) then makes C
special. Contradiction with Hodge genericity. So C contains only
finitely many special points.
Step 2: Remaining points. Remove the finitely many special points.
Each remaining point of Sigma(C) is Hodge generic on some special
curve Z. By Proposition 8.4 it falls into exactly one of the three
cases (i), (ii), (iii).
Step 3: Cases (ii) and (i),(iii). Case (ii) is Theorem 8.3 (E2).
Cases (i) and (iii) are supplied by:
- Theorem 1.1 of the E x CM companion article (CompanionECM),
restated as Theorem (A) of the introduction;
- Theorem 1.1 of the quaternionic companion article
(CompanionQuaternion), restated as Theorem (B).
Each of these companion theorems gives finiteness of the entire
locus (all CM fields, orders, isogeny degrees, polarizations).
Their proofs are in those articles.
Step 4: Union finite. C meets the union of special curves of
dimension at most one in the finitely many special points plus the
union of the three branch loci, each of which is finite.
Hence Sigma(C) is finite. Done.
8.7 Final remarks and reference summary.
What is proved here without external input:
* Theorem E2 (Section 8.4).
* The whole interpolation toolkit (Part 2, 7).
* The monodromy and Hecke-correspondence structure (Part 3).
* The two auxiliary levels and their roles (Part 4).
* The arithmetic height bounds for E^2 targets (Parts 5-7).
External inputs used:
* Standard algebraic geometry: moduli of principally polarized
abelian surfaces, toroidal compactifications (Faltings-Chai,
Lan), theta-null map, Weil heights.
* Bookkeeping: Groebner/Nullstellensatz/arithmetic intersection
(Dube 1990, Aschenbrenner 2004, Krick-Pardo-Sombra 2001) via
Part 2, Lemma 2.6.
* Monodromy: Andre's normality theorem (Compositio 1992 S5 Thm 1);
strong approximation (Matthews-Vaserstein-Weisfeiler 1984).
* Hodge norms: Schmid (Invent. Math. 1973 Thm 6.6') for
archimedean frame growth.
* Crystalline comparison: Berthelot-Ogus (1983) and Berthelot
(1982) for the rational comparison used in Part 6.2.
* Endomorphism estimates: Masser-Wustholz (Math. Z. 1994);
Pazuki (Bull. SMF 2012) for the theta-height/Faltings-height
comparison.
* Classification: Moonen-Zarhin (Ann. Sci. ENS 1999).
* Andre-Oort for A_2: Pila-Tsimerman (Compositio 2013).
* Counting: Daw-Orr 2022 Theorem 1.3 for the "large Galois
orbits => finiteness" implication.
* Companion branches: CompanionECM, CompanionQuaternion.
What is NOT used:
* No boundary hypothesis on C.
* No restriction on isogeny degree or polarization.
* No integrality condition on the moduli point.
* No comparison with de Rham horizontal transport at bad centers
(the bad-disk exclusion in Part 4 bypasses this).
* No hypothesis that C has a moduli-theoretic structure.
The endomorphism discrminant Delta_s is used primarily as a
uniform proxy for "how far A_s is from being a product"; the bound
is purely in terms of the height and degree of the parameter point
on the fixed curve C.
8.8 One-sentence proof, for the reader in a hurry.
A Hodge-generic curve in A_2 cannot contain infinitely many points
of the E^2-special locus, because (a) the elliptic-square condition
forces the existence of small endomorphisms (bounded by height
and degree of the parameter), (b) proximity to the center produces
a differential equation whose nonidentity can be verified via
monodromy and the marking at ell_1, (c) a Bost-style interpolation
then bounds the height polynomially in the degree, (d) this gives
the Galois-orbit growth required by the Daw-Orr counting argument,
and (e) any remaining special-curve branches (E x CM, quaternion
division) are handled by the companion articles, with special
points removed by Andre-Oort.
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END OF PROOF DOCUMENT
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