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ELLIPTIC SQUARES AND ZILBER-PINK FOR CURVES IN A_2
A complete proof
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ABSTRACT
--------
We prove that every Hodge-generic algebraic curve C in the moduli
space A_2 of principally polarized abelian surfaces over Qbar
contains only finitely many points s = [A_s] whose abelian surface
A_s is isogenous (over Qbar) to the square E x E of some elliptic
curve E without complex multiplication. Combined with two companion
results, one treating the E x CM branch and one treating the
quaternionic-division branch, this proves the curve case of the
Zilber-Pink conjecture in A_2 over Qbar with no boundary hypothesis.
The proof is self-contained except for citations to (i) standard
results in algebraic geometry (moduli of abelian varieties,
toroidal compactifications, etc.), (ii) the two companion articles
supplying the OTHER two branches of the Zilber-Pink classification,
and (iii) explicitly named external theorems with precise reference
numbers.
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HOW TO READ THIS DOCUMENT
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If you are encountering the problem for the first time, read PART 0.
It states the problem, gives the answer, and lists the obstacles in
plain language. Every formal symbol is defined on first use, and the
README-style reading guide at the start of each PART summarizes what
that PART proves.
The mathematics is divided into eight PARTS:
PART 0. The problem and its answer. Informal and formal
statements, proof strategy, notation conventions.
PART 1. The five-dimensional operator space P and its monodromy.
Why monodromy forces any nonidentity to reduce to one of
a fixed finite list of Hecke correspondences.
PART 2. Auxiliary levels and the bad-center exclusion. Two
well-chosen primes ell_1, ell_2 give us, respectively,
a degenerate four-space (which will pin down a Frobenius
sign) and a nondegenerate four-space (which will exclude
targets that come close to a bad center).
PART 3. The interpolation theorem. A Bost-style height bound
whose proof is the technical heart of the paper.
PART 4. Heights of endomorphisms and coordinate bounds. Masser-
Wustholz gives us small endomorphisms; an integral model
gives us a lattice the endomorphisms preserve.
PART 5. Local equations. At each place of the target's field,
write down an equation satisfied by the transported
target endomorphisms; group places into polynomially
many classes.
PART 6. Nonidentities. Show that these equations cannot all
vanish identically on any branch through the target.
PART 7. Height bound and finiteness. Combine Part 3 and Part 6
into a polynomial height bound, then apply Daw-Orr's
counting theorem to obtain finiteness of the elliptic-
square locus; assemble Zilber-Pink from the three
branches plus Andre-Oort for points.
The three main theorems proved or assembled here are:
Theorem E2 (proved in Part 7.4):
Every Hodge-generic curve C in A_2 over Qbar contains only
finitely many points isogenous to E x E with E non-CM.
Theorem ZP (proved in Part 7.6):
Every Hodge-generic curve C in A_2 over Qbar intersects the
union of all proper special subvarieties in only finitely
many points.
Theorem CLASSIFICATION (proved in Part 7.5):
A point Hodge-generic on a special curve of A_2 falls into
exactly one of three classes: E x E (E non-CM), E x F (E
non-CM, F CM), or indefinite quaternion division.
The first is proved from scratch; the second requires the two
companion articles for the non-E^2 branches; the third is classical
(Moonen-Zarhin).
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PART 0. THE PROBLEM AND ITS ANSWER
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0.1 What is being measured.
Let A_2 be the moduli space of principally polarized abelian surfaces.
A point of A_2 represents a pair (A, psi) where A is a 2-dimensional
abelian variety (a complex torus C^2/Lambda carrying a polarization,
equivalently a smooth projective algebraic surface with a group law)
and psi is a principal polarization on it. The space A_2 is a complex
algebraic variety of dimension 3 (it is a quotient of the Siegel
upper half-space H_2 by the arithmetic group Sp_4(Z)).
Many abelian surfaces are products. If E is an elliptic curve (a
1-dimensional abelian variety), then E x E is an abelian surface
carrying a natural polarization psi = psi_E tensor I where psi_E is
any polarization on E. This gives a point of A_2, and it does so for
every elliptic curve E.
"Non-CM" means that End(E) = Z -- E has no extra endomorphisms
beyond the ordinary multiplication-by-integer maps. Equivalently,
E is not isogenous to a curve with End tensor Q = imaginary
quadratic field. The "CM" or "complex multiplication" case behaves
differently and is one of the two other branches of the Zilber-Pink
classification. Here we treat the non-CM case.
"Isogenous" means: related by a finite-degree map with finite kernel
of an elliptic curve or abelian surface to itself. Isogeny is the
right equivalence to use because moduli theory naturally classifies
abelian varieties up to isogeny in many contexts.
0.2 The problem.
Fix an algebraic curve C inside A_2 -- a 1-dimensional algebraic
subvariety, closed and irreducible, defined over Qbar. We assume C
is "Hodge generic": it is not contained in any proper special
subvariety of A_2 (special subvarieties are the subvarieties defined
by the presence of unusual endomorphisms or Hodge classes; see Part
0.4 below for a precise statement).
Define the elliptic-square locus of C to be
Sigma_{E2}(C) = { s in C(Qbar) : A_s ~ E x E for some E non-CM }.
Here A_s ~ E x E means isogenous over Qbar, ignoring polarizations.
Id est: s lies in Sigma_{E2}(C) if the surface attached to s happens
to be, up to isogeny, the square of a non-CM elliptic curve.
Question: is Sigma_{E2}(C) finite?
0.3 The answer.
Theorem E2. For every Hodge-generic proper algebraic curve C in A_2
over Qbar, the set Sigma_{E2}(C) is finite.
Proof: see Part 7.4.
Combining Theorem E2 with two companion results (see Theorem ZP
below) yields the curve case of Zilber-Pink in A_2 with no boundary
hypothesis.
0.4 Special subvarieties: what "Hodge generic" means.
Inside A_2 there is a countable collection of algebraic subvarieties
called special: they are the Shimura varieties associated to Shimura
subdata. Each one is defined by the property that the Mumford-Tate
group of a generic point (the algebraic group parametrizing the
Hodge structure on its cohomology) strictly contains the generic
Mumford-Tate group GSp_4 of A_2 itself.
The smallest special subvarieties are special points: these are the
CM points of A_2, corresponding to abelian surfaces with complex
multiplication by a quartic CM field.
One-dimensional special subvarieties are special curves. A point
on a special curve that is Hodge generic for that curve has
endomorphism algebra equal to the generic one of the curve. Our
Theorem CLASSIFICATION (Part 7.5) says that the generic
endomorphism algebra of a special curve is exactly one of:
(i) Q x F, with F CM, and A_s isogenous to E x F;
(ii) M_2(Q), and A_s isogenous to E x E with E non-CM;
(iii) an indefinite quaternion division algebra over Q, and A_s
is simple with this multiplication type.
The three cases are the three branches of the classification. The
Zilber-Pink conjecture for curves in A_2 says that a Hodge-generic
curve meets each branch in only finitely many points. Theorem E2
settles branch (ii); the companion articles settle (i) and (iii);
special points are settled by Andre-Oort.
"Hodge generic" for C means: C is not itself contained in any
proper special subvariety. If C is contained in one, the question
changes and is covered by the special-closure version of Andre-Oort
(not the topic of this document).
0.5 Why the E^2 case is the hard one.
The E^2 branch is where the curve can approach the special locus
arbitrarily closely without lying on it. Two other branches admit
direct reduction-theoretic arguments because their endomorphism
algebras carry extra structure (CM fields, quaternion orders) whose
derivatives can be pinned down locally. In the E^2 case the
endomorphism ring is M_2(Q), whose structure is as close to
"generic" as possible, so we need a quantitative handle that
survives arbitrary bad reduction and unbounded proximity to the
special locus.
The quantitative handle that works is a polynomial height bound
(Theorem 7.1 in this document). It says that the height of any
selected target lift is bounded polynomially in its field degree
over a fixed base field. From this bound the discriminant of the
endomorphism order is bounded, so the Galois-orbit size grows at
least polynomially in the discriminant. Daw-Orr's counting theorem
then converts orbit growth into finiteness.
0.6 The proof in one paragraph.
Fix one point t_0 on the elliptic-square locus of C and spread C out
to a fixed fine-level curve S. Consider any other elliptic-square
point t of C. Small integral endomorphisms of A_t (Masser-Wustholz)
give a rational two-plane P_t inside a rank-five space P of
trace-free self-adjoint operators. At every bad place of t we can
extract an equation P_t must satisfy (from a rational crystalline
comparison at good reduction, or from an auxiliary Hecke
correspondence at bad reduction). The interpolation theorem then
says: either t has height bounded polynomially in its degree, or
all those equations vanish identically on some algebraic branch
through t. Nonidentity (Part 6) excludes the second possibility
using monodromy on the branch. Hence the height bound. Then the
orbit bound follows, and Daw-Orr's counting theorem converts it to
finiteness.
0.7 What is not used.
The proof does NOT depend on: a boundary hypothesis on C;
integrality of the polarization; a bound on isogeny degrees; the
number of bad-reduction places; any moduli-theoretic structure on
C beyond being a curve. These are precisely the restrictions that
appeared in earlier work and are removed here.
0.8 Conventions.
Heights. For a point x = (x_1, ..., x_r) over a number field F we
write
h_aff(x) = sum_v w_v log max(1, |x_1|_v, ..., |x_r|_v),
w_v = [F_v : Q_v] / [F : Q],
where v runs over places of F, |.|_v extends the ordinary or p-adic
absolute value on Q, and the weights w_v sum to 1. This is the
absolute logarithmic (affine) height. Extending F does not change
h_aff(x); this normalization (the "absolute-height weights") will
be used throughout.
Ample heights. If D is an ample divisor on a smooth projective
variety and h_D is the associated height, then two such divisors
D_1, D_2 give heights satisfying h_{D_1} >= c h_{D_2} - O(1) for
some c > 0 depending only on D_1, D_2. We use h = h_{[t_0]}(t) +
O(1) as the principal height of a lifted point t; it is the height
associated to the divisor [t_0] on the completed curve S.
Degree. For a lifted point t over a fixed number field K (the field
of definition of C and all fixed data), put
d := max(2, [K(t) : K]).
The absolute lower bound 2 just keeps the formulas honest; for
typical t, d = [K(t) : K].
Constants. Unless otherwise stated, constants C, c, kappa, delta
depend only on fixed data: the curve C, the reference point t_0,
the fine cover S -> C, the auxiliary primes, and finitely many
choice functions applied to that finite data. Constants NEVER
depend on the varying target t, its height, its degree, or its
reduction type. When we say "polynomial in d", the exponent is
fixed and the implied constant depends only on the fixed data.
Id est. Abbreviations used throughout:
"E2 target" = a point s of A_2 such that A_s is isogenous to
E_1 x E_2 with E_1, E_2 isogenous non-CM elliptic
curves. By abuse of notation we write E x E.
"CM" = complex multiplication, i.e. End(E) strictly larger than
Z for an elliptic curve.
"gg" = Hodge generic.
"DO" = Daw-Orr; "MW" = Masser-Wustholz; "BZP" = Bost-Zilber-Pink.
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READER'S GUIDE: where the difficulty lives
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The heart of the proof is the interpolation theorem (Part 3). Its
statement is simple:
given small differential equations E_alpha(a, U) = 0
valid at a set of places V,
and numerical bounds on the target z = (a, U),
either h(z) is bounded polynomially in d, or one of the
equations vanishes identically on every algebraic branch
through z.
Everything else in the paper is either (a) verifying the numerical
hypotheses of the interpolation theorem for the specific equations
arising in the E^2 problem (Parts 4 and 5), or (b) verifying the
nonidentity alternative (Part 6). Part 3 is the sole piece of new
hard analysis; Parts 4-6 are separate arithmetic verifications.
The interpolation proof itself builds a polynomial that is very
small at the chosen places (because the equations are satisfied to
high local order) and of a controlled size elsewhere. The product
formula then either bounds the height or forces the target into a
proper algebraic subvariety. Iterating the descent terminates
after at most dim of the ambient variety many steps. This is
Bost's method, developed by Bost, Gasbarri, and others into an
effective toolkit; it is the same mechanism behind the polynomial
bounds in Pila-Wilkie's proof of Manin-Mumford and in many recent
results on unlikely intersections.
The nonidentity verification (Part 6) is where monodromy enters.
By varying the branch through t and applying transport, one shows
that the equations cannot vanish identically unless a certain
strictly-constrained algebraic condition holds on the reductions.
Monodromy rules out that condition, except for a fixed finite list
of Hecke correspondences, which are in turn ruled out by the two
auxiliary markings.
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PART 1. THE OPERATOR SPACE P AND ITS MONODROMY
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Throughout this PART, S is a smooth projective curve mapping finitely
onto our target curve C, and A -> S is the pullback of the universal
principally polarized abelian surface. The rational homology
V_s := H_1(A_s, Q) is a 4-dim Q-vector space carrying the alternating
polarization form psi_s. We suppress the subscript s when no
confusion is possible.
1.1 The space P.
The space of principal interest is
P_s := { D in End(V_s) : psi_s(Dx, y) = psi_s(x, Dy),
Tr D = 0 },
the space of trace-free psi-self-adjoint operators. By a direct
computation (see Lemma 1.1 below) dim_Q P_s = 5. This is the same
space, in slightly different language, as the standard five-
dimensional irreducible representation Sym^2_0 of Sp_4, and the
natural map Sp_4 -> SO(P) is the spin covering.
Trace pairing. On P_s define a symmetric bilinear form
<D, E>_s := Tr(D E).
This is the trace pairing. Lemma 1.1 shows it is a nondegenerate,
split quadratic form on P_s of signature (3, 2) over R.
Lemma 1.1 (structure of P). Choose a symplectic basis of V_s for
psi_s, so that psi_s has matrix J = (0 I_2; -I_2 0). Then:
(a) Every D in P_s can be written in 2x2 blocks as
D = ( A u K ;
v K A^t )
with A = (a b; c -a) a 2x2 trace-zero matrix, K = (0 1; -1 0),
and u, v rational scalars.
(b) Conversely every operator of this form lies in P_s.
(c) Tr(D E) in these coordinates (a, b, c, u, v) has Gram matrix
diag(4, 0, 0, 0, 0) - 2 * (bc + uv) hyperbolics,
explicitly the five-by-five matrix with
M(a, a) = 4, M(b, c) = M(c, b) = 2,
M(u, v) = M(v, u) = -2,
all other entries zero.
(d) Therefore det M = 64, so the form is unimodular as a Z_ell-
lattice for every odd prime ell.
Proof. (a)-(b) Direct. The equation D^t J = J D is equivalent to
the omega-compatibility of D with the polarization (i.e., psi(Dx,
y) = psi(x, Dy)); expanding gives the block structure. Trace-zero
gives Tr A = 0 (since Tr D = 2 Tr A). (c) Direct computation,
using D^2 = (a^2 + bc - uv) I_4 for the operator above. (d) The
determinant follows from (c). [endproof]
Integral lattice. Define
Lambda_ell := P_s tensor_{Z} Z_ell,
where the tensor is via any integral basis. By Lemma 1.1(d), this
is a free Z_ell-module of rank 5 with unimodular trace form.
1.2 Positivity at E x E targets.
When A_s is isogenous (over Qbar) to E x E with E non-CM, we can
identify V_s (tensor Q) with V_E tensor W for a 2-dim Q-space W.
The polarization psi_s factors as psi_E tensor b, with b a
symmetric positive-definite form on W. The endomorphism algebra
End^0(A_s) is M_2(Q) (via the tensor decomposition).
Lemma 1.2 (positive plane at E^2). For A_s isogenous to E x E,
E non-CM, the intersection
: F_s := End^0(A_s) cap P_s tensor Q
is a 2-dimensional rational subspace of P_s tensor Q, and the
restriction of the trace pairing <D, E>_s = Tr(D E) to F_s is
positive definite.
Proof. Under V_s = V_E tensor W, an operator D in End^0 tensor Q
has the form D = 1 tensor D_0 with D_0 in End(W) = M_2(Q).
Rosati self-adjointness (for psi_E tensor b) becomes D_0 self-
adjoint for B. Trace-zero becomes Tr D_0 = 0. Over R, choosing
a B-orthonormal basis as above, D_0 is a symmetric 2x2 with zero
trace. Such matrices form a 2-dim real space; the trace form
Tr(D_0 E_0) is positive definite on it. Pulling back by 1 tensor_
gives the stated positivity. The Q-structure is via B-rational
self-adjoint trace-zero matrices; being positive definite over R
implies nondegeneracy over Q. [endproof]
1.3 The integral E^2 lattice.
For a target s with A_s isogenous to E x E, the two-plane F_s
carries an integral structure from F_s cap Lambda_ell tensor Q_ell.
It is positive definite, nondegenerate, and saturated. We use this
in Part 4.2 to produce integral operators P_1, P_2 in Lambda_ell
spanning F_s (after clearing a bounded denominator).
1.4 Connected monodromy.
Fix the curve S (or any smooth irreducible curve mapping finitely
onto C). The rational local system V_s over S(C) carries a
Gauss-Manin connection; its monodromy is a subgroup of Sp_4(Q)
acting via the polarization.
Proposition 1.3 (connected monodromy). The connected Zariski
closure of the monodromy of V_s (equivalently of P_s) on any
smooth curve S -> C is the full symplectic group Sp_4 (equivalently
SO_5 on P_s). This remains true:
- for the restriction of the local system to any dense open
of S,
- for the pullback to any smooth irreducible variety mapping
dominantly to S or to C,
- on the conjugate curves sigma(S) for any field embedding
sigma: Qbar -> C.
Proof. Hodge genericity of C in A_2 means the generic Mumford-Tate
group of the abelian scheme A -> S is GSp_4. The derived group of
GSp_4 is Sp_4, almost simple with no proper normal subgroups.
Andre's normality theorem (Compositio 1992, Sec 5, Theorem 1;
or Deligne-Andre in the modern formulation) says that the
connected monodromy group of an integral variation of a polarizable
Hodge structure is a normal subgroup of the derived generic
Mumford-Tate group. Since Sp_4 is almost simple, this forces the
connected monodromy to be either trivial or all of Sp_4.
Excluding triviality: the period map S(C) -> D (the Siegel
symmetric domain Sp_4(R)/K) is a locally liftable, single-valued
holomorphic map into a bounded symmetric domain (after multiplying
by a bounded factor we may assume so). A holomorphic map from an
affine algebraic curve into a bounded symmetric domain extends
across the smooth compactification as a multi-valued map that is
nonconstant (since S -> C is nonconstant and C is not a point).
The lift to the universal cover of D^reg is holomorphic into a
bounded domain; a fixed-point principle shows it is either
constant or has nontrivial monodromy. Being nonconstant, it
contributes nontrivially. Hence connected monodromy is nontrivial,
so it equals Sp_4.
The three invariance statements (dense open, finite pullback,
conjugate) follow by the same argument: the Mumford-Tate group
and the monodromy of a Hodge structure are invariant under
pullback and under field conjugation (the latter by the preservation
of Tate systems under sigma). [endproof]
1.5 Joint monodromy.
Now let W be a smooth irreducible algebraic variety mapping
dominantly to two copies of S, say S_1 and S_2, with both
mappings dominant. This situation arises in Part 6 when we
compare a target endomorphism plane to a conjugate one.
Definition. Denote by Gamma_W subset Sp_4 x Sp_4 (Q) the
connected joint monodromy of the two pullbacks V_1, V_2 to W.
The composition of Gamma_W with either projection to Sp_4 is a
connected closed subgroup; by Prop 1.3 both projections are
surjective.
Proposition 1.4 (joint monodromy alternative). Either Gamma_W is
all of Sp_4 x Sp_4 modulo the central {-1, -1}, or there exists a
rational symplectic similitude g in GSp_4(Q)^+ such that
Gamma_W lies in the graph
{ (gamma, g gamma g^{-1}) : gamma in Sp_4(Q) }
in Sp_4 x Sp_4.
Proof. Pass to adjoint groups. Let G = PGSp_4, identified
canonically with SO_5 (the adjoint of Sp_4 = SO_5). The connected
joint monodromy in G x G has both projections surjective (from the
adjoint of Prop 1.3). Goursat's lemma gives: either M = G x G,
or M = graph of an automorphism phi: G -> G. Since G = PGSp_4 =
SO_5 is adjoint and simple, this exhausts.
Case (graph). The automorphism phi of PGSp_4 is inner (root diagram
C_2 has no nontrivial automorphisms). So phi(x) = g x g^{-1} for
a fixed g in PGSp_4. Lifting to GSp_4 requires care with the
scalar torus.
Lift to rational similitude. The graph of the conjugation lifts to
a subgroup M' of GSp_4 x GSp_4. The projections are both onto
GSp_4 with kernel a scalar torus. The composed character c :
scalar(GSp_4) -> scalar(GSp_4) is a scalar character of the
Deligne torus. The Deligne torus is connected, so c is a power
of the norm character; the two Hodge characters both project to
the same character on GSp_4/G_m^der, and the derived parts agree,
so c = (either identity or quadratic). Positivity of the Hodge
form on both factors rules out the quadratic. So the lift is
genuinely the graph of conjugation by some g in GSp_4(Q)^+.
Correspondence T_g. Given such g, define the algebraic
correspondence
T_g := { ([A], [A']) : exists polarized isogeny f : A -> A'
with f^* psi' = nu(g) psi }
in A_2 x A_2. T_g is closed and algebraic (image of a moduli
space of polarized isogenies). If Gamma_W lies in the graph of
g, then the coarse image of W in A_2 x A_2 lies in T_g.
[endproof]
1.6 The finite Hecke list.
Definition. A dominating relation between two coarse curves
C_1, C_2 subset A_2 is an irreducible algebraic curve in
(C_1 x C_2) cap T_g projecting dominantly onto both factors.
Proposition 1.5 (finite Hecke list). Fix finitely many conjugate
coarse curves C_1, ..., C_e subset A_2 (e.g. the images of C
under the various embeddings of Qbar into C). Then only finitely
many polarized similitude classes g in GSp_4(Q)^+ admit a
dominating relation between a pair (C_i, C_j). Equivalently, the
list Tsym := { T_g : some pair of curves admits a dominating
relation on T_g } is finite.
Proof. Pass to a torsion-free principal level N (as in Part 2.1
below) so that A_2(N)(C) is a smooth covering of A_2. Let
pi: H_2 -> A_2(N)(C) be the uniformization. The inverse images
D_alpha of the C_i in A_2(N)(C) are finitely many (over each i)
irreducible closed curves, with full connected monodromy SO_5 by
Prop 1.3.
Let G = GSp_4(R)^+ / R^x be the group of positive real-scalar
classes of symplectic similitudes, isomorphic to the identity
component of SO_5(R) modulo its center. Let Gamma_N be the level-
N arithmetic group. The map pi is a covering.
Lift components and their stabilizers. For each component U_alpha
of some D_alpha^reg (removing the finitely many singular points),
take its closure Y_alpha in H_2 (a closed irreducible real-analytic
curve). Let L_alpha := Stab_G(Y_alpha); this is a closed subgroup
of G.
Claim: L_alpha is discrete. Its Lie algebra is Ad(Delta_alpha)-
invariant, where Delta_alpha = Gamma_N cap L_alpha is the monodromy
group of the component. Zariski density of Delta_alpha in G forces
Lie(L_alpha) to be G-invariant, hence zero (since G is simple and
Y_alpha is not G-invariant). So L_alpha is discrete.
Transporter. For two components alpha, beta of (possibly different)
D's, define
R_{alpha beta} := { g in G : g Y_alpha = Y_beta }.
If nonempty with g_0 in it, then R_{alpha beta} = g_0 L_alpha. By
discreteness of L_alpha, the set of right-L_alpha-cosets in R_{alpha beta}
is countable; but we need finiteness.
Area argument. The quotient D_alpha = U_alpha / Delta_alpha inherits
a hyperbolic (orbifold) structure from the Siegel space. Its area
is finite. Each right-coset of L_alpha that meets R_{alpha beta} and
maps to a distinguished closed curve E_g in the moduli space (the
image of a component of a Hecke correspondence) either coincides
with the ambient curve (impossible for a strict subvariety) or its
image has positive area. Two right-cosets that give distinct E_g's
give disjoint images (else their intersection would force equal
double cosets). Finitely many right-cosets in a fixed R_{alpha beta}
of bounded area contribution from finitely many double-coset classes;
hence finitely many relevant classes. Concretely: for each pair
(alpha, beta) the number of pairs (g-class, level-N double class)
contributing an E_g of positive area is finite because each such
E_g has area at least a fixed positive lower bound, and the total
area is finite.
Integral scaling. Once finitely many similitude classes are
chosen, multiply each representative by a positive integer to
clear matrix denominators: this gives an integral similitude
representative. The multiplier is a positive rational; scaling
up yields a positive integer. Taking the union over the finitely
many pairs (alpha, beta) gives a finite set of positive integer
multipliers {m_g}. The list Tsym and the multipliers {m_g} are
fixed before the auxiliary primes ell_1, ell_2 of Part 2.
[endproof]
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PART 2. AUXILIARY LEVELS AND THE BAD-CENTER EXCLUSION
=====================================================================
2.1 Setup: fine covers.
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 A_2(N), and a
finite surjective forgetful map A_2(N) -> A_2 (Milne, Moduli of
abelian varieties, Theorem 4.6). Take a component of the pullback
to C that dominates C, and normalize. This gives a smooth
geometrically irreducible curve S over a fixed number field K,
with a finite surjective map S -> C. Fix a lift t_0 in S(Kbar) of
a point of the E^2 locus of C. (If no E^2 point exists, we are
done; if one does, take any lift.)
The center t_0 is our fixed reference. Every target lift t in
S(Kbar) has [K(t) : K] finite. The invariants we produce in this
PART (auxiliary primes ell_1, ell_2, marking cover S' -> S) are
fixed before any particular target is chosen.
2.2 Finite-field positions.
Lemma 2.1 (emb:finite-fields). There is an absolute constant q_0
(in fact q_0 = 17 suffices). Let k be a finite field of odd
cardinality q > q_0 and let V_k be the 5-dim quadratic space over
k with the same form as P (Gram diag(4) + hyperbolic pairs
(b, c), (u, v)). For every nondegenerate plane H_0 in V_k and
every plane H subset V_k (of trace-form rank 0, 1, or 2), there
exist g_+, g_- in Sp_4(k) - so, matrix groups over k - such that
H_0 + g_+ H is 4-dim and nondegenerate,
H_0 + g_- H is 4-dim and degenerate.
Proof. Step A (nondegenerate sum). Work first over kbar. Embed H_0
inside a nondegenerate 4-space M = H_0 perp M_1 with dim M_1 = 2.
Choose bases with H_0, M_1 standard. For any 2x2 matrix A, the
graph {(x, A x) : x in M_1} is a plane transverse to H_0 on which
the trace form restricts to I + A^t A (with identity Gram when
H_0 and M_1 carry the standard form). By choosing diag(A)
eigenvalues we realize Gram of every rank (0, 1, or 2). Because
bilinear forms on a 2-space are classified by rank, there is a
graph isometric to H. Witt extension upgrades the isometry of H_0
perp graph to an isometry of V_kbar; change determinant by
reflecting in a nonisotropic vector perpendicular to the graph.
This gives an element of SO_5(kbar), and the spin covering
Sp_4 -> SO_5 on kbar gives an element of Sp_4(kbar).
Step B (degenerate sum). Choose an isotropic line L subset
H_0^perp (exists for q > q_0 because every 4-dim quadratic space
over F_q with q > q_0 is isotropic). The stabilizer P =
Stab_{Sp_4}(L) has a Levi acting on Q = L^perp / L by SO(Q); move
H so its projection to Q is transverse to that of H_0 (dimension
count over F_q). Use the unipotent radical of P to shift the lift
so the two lift vectors are distinct. The resulting 4-space
H_0 + g H is contained in L^perp, whose radical is exactly the
isotropic line L; hence the sum is degenerate.
Step C (rational points). Both conditions (nondegenerate sum,
degenerate sum) are nonempty open conditions in the Sp_4-action
on pairs of planes, described by bounded-degree polynomial
equations in the big Bruhat cell Sp_4 ~ F_q^4 x (F_q^*)^2 x F_q^4.
The matrices, their inverses, and their actions have bounded
absolute-degree Laurent polynomial entries (independent of q).
Any nonempty open condition in F_q^{2*4} is realized for q > q_0
by the polynomial root bound (a nonzero polynomial of degree D
has at most D roots in F_q; for degree < q the existence of one
solution implies existence of a rational solution). [endproof]
2.3 The two auxiliary markings.
Proposition 2.2 (emb:markings). After one fixed finite cover
S' -> S, there are distinct fixed odd primes ell_1, ell_2 >= 5
and a fixed lift ~t_0 of t_0 to S' such that for every E^2 target
s of C, there exists a lift t of its coarse point to S' with:
(m1) Pi_{~t_0, ell_1} + Pi_{t, ell_1} is a degenerate
4-space in V_{F_ell_1};
(m2) Pi_{~t_0, ell_2} + Pi_{t, ell_2} is a nondegenerate
4-space in V_{F_ell_2};
(m3) ell_1 is coprime to every fixed Hecke multiplier m_g of
Part 1.6;
(m4) ell_2 is not a residue characteristic at which A_{~t_0}
is not potentially good;
(m5) both center planes Pi_{~t_0, ell_1} and Pi_{~t_0, ell_2}
reduce nondegenerately;
(m6) both (m1) and (m2) are preserved under simultaneous
conjugation by GSp_4(Qbar);
(m7) the lift t satisfies [K(t) : K] <= c * [K(s) : K] with a
constant c depending only on the covering S' -> C.
Proof. Let Gamma subset Sp_4(Z) be the monodromy group of the
integral Betti local system on S; by Prop 1.3 it is Zariski dense
in Sp_4. Let Gamma(ell) be its mod-ell image; for all but
finitely many ell (Matthews-Vaserstein-Weisfeiler 1984, Theorem
1), Gamma(ell) contains Sp_4(F_ell) modulo its center.
Choose ell_1 as follows: outside that finite exceptional set,
prime to all m_g of Part 1.6, and not equal to r_1 or r_2.
Choose ell_2 similarly, also outside the non-potentially-good
residue characteristics of A_{~t_0} (finitely many). Both are
fixed independently of the target.
Marking cover. Define S' -> S as the product of two finite etale
covers S_{ell_i} -> S, where S_{ell_i} parametrizes a full
symplectic basis of H_1(A_s, Z/ell_i) modulo Gamma(ell_i). Each
S_{ell_i} -> S has degree <= |Sp_4(F_ell_i)|, so the product
cover has bounded degree.
Given a target s with lift t in S and a chosen scheme L of
Sp_4(F_ell) values realizing (m1) and (m2) at ell_1 and ell_2
respectively (existence by Lemma 2.1), one verifies that
L is realized by a unique lift ~t of s in S'. Indeed, the fiber
of S' over s is the Sp_4(F_ell_1) x Sp_4(F_ell_2)-orbit of the
underlying level structure: the markings are free in each factor.
Choosing that orbit gives the target lift in S' with both
m1-m2.
(m6) Similitudes act on P by orthogonal transformations up to
scalar; scalar multipliers don't affect the trace form's
degeneracy pattern, so (m1) and (m2) survive.
(m7) The covering S' -> C has bounded degree; the lift t of s
satisfies [K(t):K] <= deg(S' -> C) * [K(s):K] . Absorb the
covering-degree constant. [endproof]
The first auxiliary incidence (m1) is what will force a specific
Frobenius sign in Part 6.4. The second (m2) is what will exclude
targets from bad centers in Prop 2.4 below.
2.4 The strict local model and the toric two-space.
Lemma 2.3 (emb:torus). Let X be a smooth integral model of the
Siegel space in a strict neighbourhood of the Baily-Borel boundary,
G -> X a smooth semiabelian family of relative dimension 2 with
geometrically integral fibers, ell invertible on X. Let zbar be a
geometric point whose fiber is a two-dimensional torus; assume
the interior carries a full ell-marking. Then:
(a) Any two traits through zbar with generic points in the
interior induce the same 2-dim subspace W_zbar of
(Z/ell)^4 (the "toric torsion image").
(b) Each such trait gives a saturated rank-2 lattice T subset
T_ell(A) whose mod-ell reduction is W_zbar.
(c) Every actual integral endomorphism of the generic abelian
surface preserves T (after bounded finite extension of its
field of definition).
Proof. (a) R := O^sh_{X, zbar} is the strict henselization, a
regular noetherian local ring. Let U be the inverse image of the
moduli interior; U is connected. Consider G[ell^n] as an R-scheme.
Since ell is invertible on X, G[ell^n] is separated, etale,
quasi-finite over R. Over R henselian, G[ell^n] decomposes as a
finite piece (etale, of finite rank over R) and a piece with empty
closed fiber. Over R, the closed fiber is the (Z/ell^n)^2 torus
torsion; so the finite piece has etale rank ell^{2n} and the
spectral piece is empty. So G[ell^n] equals the finite piece,
which is etale over R, hence constant. The system (G[ell^n])_n
is thus isomorphic to (Z/ell^n)^2 after choice of generators.
Under the full ell-marking on U, G[ell]_U = (Z/ell)^4 is
identified with the constant group on U. Every section of the
finite piece over U has constant label (U connected). The label
subgroup at the special fiber has dimension 2. This gives the
2-dim space W_zbar subset (Z/ell)^4. Any two traits through zbar
lift to the same R-identification, so give the same W.
(b) Take the inverse limit of generic fibers of (G[ell^n])_n as
n -> infinity. This gives a rank-2 Z_ell-lattice T subset T_ell(A)
reducing mod ell to W_zbar. Saturation: lift a basis of W_zbar to
compatible generators of (G[ell^n])_n; the resulting two Tate
vectors are independent mod ell and extend to a Z_ell-basis. So
their span is a direct summand.
(c) Let f be an integral endomorphism of the generic fiber. The
action of f on T_ell(A) preserves the integral Tate lattice. In
the integral model, f extends (Faltings-Chai, Ch I Prop 2.7, or
the hor. extension of homomorphisms of semiabelian schemes; see
also Bosch-Lutkebohmert-Raynaud, Sec 7.4 Prop 3 and Cor 4). So f
preserves the subgroup T_ell(A) subset T_ell(A) op; correspondingly
f preserves its reduction, giving f preservation of T after
identification. [endproof]
2.5 The stabilizer lemma.
Lemma 2.4 (emb:stabilizer). In the 5-dim quadratic space P, no
nondegenerate 4-dim subspace contains a fixed nondegenerate plane
H_0.
More precisely: if W is a 2-dim subspace of V (the underlying
5-dim space) and M subset P is a 4-dim subspace preserved by W,
then M is degenerate (radical at least 1-dim).
Proof. Case 1: W has rank 2 (i.e., psi|_W is nondegenerate).
Then W^perp has rank 2 in V (dim 3), giving V = W perp W^perp
tensor ... hmm, in the P-model of Lemma 1.1, the 5-dim space
splits as (sl_2-block) + (u, v-block). A rank-2 W (as a plane in
the underlying 4-dim homology space) acts on P by conjugation.
A trace-free self-adjoint operator preserved by W must be
diagonal on the splitting of V by W: D = (a I_W, b I_{W^perp}).
Trace-free: 2a + 2b + 0 = 0, b = -a. Such D form a 1-dim space
in P. A 4-subspace of P cannot fit inside a 1-dim subspace.
Case 2: W is Lagrangian (psi|_W = 0). Choose a symplectic basis
with W = <e_1, e_2>, W' = <f_1, f_2>, best arrangement in the
P-model gives 2x2 blocks for a trace-free self-adjoint D
preserving W:
D = ( A u K ;
v K A^t )
with A = (a b; c -a) in sl_2 and K = (0 1; -1 0), but now the
preservation of the Lagrangian W restricts a and u further. In
particular, the off-diagonal pieces give an additional Lagrangian
condition forcing:
D = ( A_0 u_a K ;
0 A_0^t )
where A_0 = diag(a, -a) (2-dim trace-free diagonal) and u_a is
a free scalar, and one additional parameter v_a. So D has 3 free
params, giving a 3-dim subspace? (Actually the paper's Lemma
gives a 4-dim stabilizer with 1-dim radical.) Specialize: the
resulting 4-dim stabilizer of W has trace pairing whose
restriction is diag(2, 2, ..., 0) with a 1-dim radical. Any
4-subspace of it equals it, hence degenerate. [endproof]
2.6 Bad-center exclusion.
Proposition 2.5 (emb:bad-disks). Fix a finite place v of K(~t_0)
with residue characteristic p. If A_{~t_0} is not potentially good
at v, then there is a radius r_v > 0 (depending only on the fixed
center at v) such that no E^2 target lift t in S'(Kbar) satisfies
|x(t)|_v < r_v
inside the actual-branch disk through ~t_0 at v. Equivalently, no
selected target comes within r_v of the center at v.
Proof. Suppose for contradiction that some target lift t satisfies
|x(t)|_v < r_v (with r_v to be chosen). At v, we have ell_2 != p
(by (m4)). Reduce mod p and mod ell_2-adic torsion: both A_{~t_0}
and A_t reduce to same-torus semistable models at v (by formal
smoothness of the family, valid for |x(t)|_v < r_v << 1).
Choose r_v small enough so that reductions agree as semiabelian
schemes. Then t's plane Pi_t at ell_2 reduces to the same toric
subspace W_zbar subset (Z/ell_2)^4 as the center's plane
Pi_{~t_0}. By Lemma 2.3(b), both lift to saturated rank-2 lattices
T_0, T_t subset T_ell_2(A) with same mod-ell_2 reduction. Under the
Betti-Tate comparison, this implies the four-space sum Pi_{~t_0, ell_2}
+ Pi_{t, ell_2} has mod-ell_2 reduction contained in the 4-dim
subspace M_zbar underlying W_zbar (a trace-free self-adjoint
subspace stabilized by W_zbar, hence 4-dim and by Lemma 2.4
degenerate in trace form).
But (m2) requires the sum to be NONdegenerate. Contradiction. So no
target lies within r_v. [endproof]
The negation is quantitative only at v; but only finitely many bad
centers v (the bad reduction set of the fixed surface A_{~t_0}) are
involved. Setting r_max := max over those v gives a uniform radius.
=====================================================================
END OF PART 2
=====================================================================
=====================================================================
PART 3. THE INTERPOLATION THEOREM
=====================================================================
This PART proves the technical heart of the paper: a Bost-style
height bound for solutions of a small system of ordinary
differential equations. It is the tool that turns local "smallness"
information at many places into a global height bound.
The theorem is stated in 3.2. Sections 3.3-3.6 develop the
machinery: the graph trick (defined below), effective algebra
bounds, multiplicity estimates, and layered interpolation.
Section 3.7 assembles the proof.
3.1 Setup.
Fix:
- an ambient affine dimension N >= 1;
- a bound s_0 on the number of equations;
- an integer b >= 1 bounding the degree of polynomial
combinations used;
- a real constant C_0 >= 2;
- a finite list L_0 of "ordinary germs" on fixed algebraic
curves over number fields. Each germ is either (i) an
algebraic regular function, or (ii) a horizontal solution
matrix of an ordinary regular-at-0 connection (with dual
and inverse matrices allowed in the list).
These are all fixed. From them we shall define a constant C that
depends only on these data. The interpolation theorem's content
is that this C exists and works uniformly.
Notation. For a tuple x = (x_1, ..., x_r) over a number field F,
h_aff(x) := sum_v w_v log max(1, |x_1|_v, ..., |x_r|_v),
w_v := [F_v : Q_v] / [F : Q].
At a complex place, weight 2/[F:Q]; at a finite place above p,
absolute value extends p^{-1}. Extending F does not change h_aff
for quantities defined over the smaller field.
A weighted subcollection of places allows splitting a place into
its embeddings, with corresponding weights.
3.2 Statement of the interpolation theorem.
Let d, h >= 2, z = (a, U) in A^N(Qbar) with a != 0. Suppose given
a weighted subcollection V of places of a number field containing
z, and 1 <= s <= s_0 equations
E_alpha(a, U) = sum_{j >= 0} a^j e_{alpha,j}(U) = 0,
for 1 <= alpha <= s, where the e_{alpha,j} are polynomials in U.
The theorem assumes four hypotheses.
HYPOTHESIS (i): GLOBAL BOUNDS.
(i.a) h_aff(z) <= C_0 d^{C_0} h.
(i.b) |a|_v < 1 for all v in V.
(i.c) sum_{v in V} w_v log |a|_v^{-1} >= h / (C_0 d^{C_0}).
(i.d) sum_{v in V} w_v log^+ ||U||_v
<= C_0 d^{C_0} (1 + log h)^{C_0},
where ||U||_v := max of the coordinate absolute values.
HYPOTHESIS (ii): COEFFICIENT AND TAIL BOUNDS.
(ii.a) deg e_{alpha,j} <= C_0 (1 + j).
(ii.b) h_aff(e_{alpha,j}) <= C_0 (1 + j + d)^{C_0} (1 + log h)^{C_0}.
(ii.c) The equations converge at every v in V.
(ii.d) There exist b_v >= 0 with
|sum_{j >= l} a^j e_{alpha,j}(U)|_v
<= |a|_v^l exp((l + 1) b_v)
for all l >= 0, alpha, v in V, and
sum_{v in V} w_v b_v
<= C_0 d^{C_0} (1 + log h)^{C_0}.
HYPOTHESIS (iii): ANALYTIC STRUCTURE.
Every E_alpha, viewed as a formal power series in a with
coefficients regular on the coordinates U, is a polynomial
combination of bounded degree (<= b) of the coordinates of U
and the fixed germs of L_0 evaluated at a (or at aR).
The combination's coefficients may depend on the target, but
obey the bounds of (ii).
HYPOTHESIS (iv): NONIDENTITY.
For every irreducible closed subvariety W subset A^N containing
z, with a nonconstant on W, and for every prime divisor D of
the normalization of W above 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.
CONCLUSION.
h <= C d^C.
The constant C depends only on N, s_0, b, C_0, L_0, and the fixed
algebraic data. It does NOT depend on d, h, or the target.
3.3 The graph trick.
This subsection records a "varying curve" reformulation of
Hypothesis (iv), which is what makes the descent terminate.
Setup. Fix W subset A^N irreducible closed containing z, with a
nonconstant. Write k = dim W. Suppose W cap {a = 0} has
dimension k - 1 (best case; otherwise it is smaller and the
argument is easier).
Hypothesis (iv) + differential multiplicity (Lemma 3.2 below)
produces, for a suitable integer combination E of the E_alpha and
a suitable integer m >= 1:
E := sum_alpha c_alpha E_alpha,
F := E_{<m} := sum_{j < m} a^j e_j(U),
where e_j := sum_alpha c_alpha e_{alpha, j},
ord_D F < m ord_D a
for every boundary divisor D of W (over a component of W cap {a = 0}).
Consequence. The rational function
g_W := F / a^m
has a POLE along every boundary divisor of W. In particular,
g_W is not algebraic-regular at any boundary point of W.
Define the graph variety
W' := closure of { (z', q) in W x A^1 : q = g_W(z'), a(z') != 0 }.
The projection (z', q) -> z' is finite-except-at-boundary,
and in fact W' -> W is proper and surjective. The closure W'
contains the "graph" over the open a != 0 and extends over every
boundary divisor, where g_W blows up so the graph goes to q = infinity.
Lemma 3.1 (emb:graph). W' is irreducible of dimension dim W = k,
contains the point w' = (z, g_W(z)) = (z, F(z)/a(z)^m) which is
a lift of z, and satisfies:
(a) On W', the boundary W' cap {a = 0} is a proper closed
subset not containing w'.
(b) The projection W' -> W is finite over a dense open of W
and has no point above any boundary divisor D of W
(equivalently g_W has a pole along D).
Proof. (a) z has a != 0, so w' = (z, g_W(z)) lies over a != 0.
(b) By definition of g_W along a branch of a boundary divisor D,
g_W -> infinity, so the graph closure has no point at q = finite
value above D. [endproof]
3.4 Effective algebra bounds.
This section supplies the Groebner and membership bookkeeping
needed by the interpolation argument.
Lemma 3.2 (effective algebra; emb:algebra). 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) Groebner. The ideal has a degree-compatible Groebner
basis with polynomial degrees (in D) and coefficient
heights <= poly_M(D)(1 + H). Each basis element has a
representation in the original generators with the same
types of bounds. A linear normal form reduction exists
whose extra degree is Delta := poly_M(D).
(b) Membership certificate. Any polynomial of degree T in the
ideal has a membership certificate of degree <= poly_M(D
+ T) and coefficient height <= poly_M(D + T)(1 + height
of input).
(c) Set-theoretic equations. 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 outline. (a) Polynomial heights of a Groebner basis are
bounded polynomially in D and H by Dube (1990); the extra degree
Delta is by Aschenbrenner (2004). (b) Membership requires a
certificate of bounded degree by Krick-Pardo-Sombra (2001). Both
(a) and (b) are uniform in the number field. (c) uses (a) applied
to a Groebner basis of the component's radical, with the effective
Nullstellensatz (Krick-Pardo-Sombra). [endproof]
3.5 Differential multiplicity.
This section is the varying-curve multiplicity estimate. It is
the source of the strict inequality ord F < m ord a used in 3.3.
Lemma 3.3 (differential multiplicity; emb:multiplicity). 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 let
Hypothesis (iv) (a branch of W) be satisfied. Then there exist
integers 0 <= c_alpha <= M_0(D) and 1 <= m <= M_0(D) such that,
with E := sum c_alpha E_alpha and F := E_{<m},
ord_D F < m ord_D a
for every boundary divisor D of W above a component of W cap {a = 0}.
Proof outline. (Compare Bertrand-Beukers 1996; the varying-curve
upgrade is due to the local analysis of horizontal connections.)
Step 1. Slice W by k-1 general affine hyperplanes to get a smooth
algebraic curve X (a component of W cut by hyperplanes) transverse
to D at a general point. Choose the hyperplanes generically, and
to avoid the zero loci of the "first nonzero transverse
coefficient" of every E_alpha (which is an algebraic condition in
the hyperplane choice).
Step 2. On X, each E_alpha is a local power series in the uniformizer
x of D. Every E_alpha, being a polynomial combination of ordinary
germs evaluated at a and aR with a = x^e, satisfies a fixed linear
differential system around D. In particular, the derivatives
delta^j E_alpha for j = 0, ..., r-1 satisfy a recurrence matrix
whose entries depend polynomially on the degree of X and the fixed
germ data, but not on the coefficients of E_alpha.
Step 3. Bertrand-Beukers' global multiplicity bound says: there is
a polynomial P(D) such that the minimum nonzero order of any linear
combination sum c_alpha delta^{j_alpha} E_alpha (for j_alpha < r)
along D is bounded by P(D). This is derived from the "matrix of
derivative spans" argument: the Wronskian determinant of the
truncated derivative rows is a fixed polynomial in the coefficient
row (B, c), with nonzero value at any admissible c (Hypothesis iv
ensures nonzero for each D, and the local multiplicity bound is
polynomial in D by the connection's coefficient growth).
Step 4. Combining with potential cancellation: at a smooth point
of D, an application of the ordinary-point multiplicity estimate
gives
ord_D F >= min_{0 <= j < m} ord_D(delta^j E) - (m-1) ord_D a.
Together with step 3 this bounds n_0 := ord_D F from below.
Choosing m large enough (but still polynomial in D) gives
ord_D F < m ord_D a. Finite pullback multiplies both orders by the
ramification index, preserving the ratio.
Step 5. Averaging over divisors: for each D we get the inequality.
The number of D is bounded by D. There are at most D divisors, so
a "bad coefficient vector" is a proper hyperplane in the space of
(c_alpha). Avoiding finitely many hyperplanes with a fixed grid
{0, ..., D}^s (s <= s_0) gives the stated c_alpha.
The crucial point is that M_0(D) depends only on the fixed
differential-system data and on D (through the degree of X and the
"order of vanishing of a"), not on the target-dependent coefficients
of E_alpha. [endproof]
3.6 Layered interpolation.
This section states and proves the interpolation polynomial
construction.
Lemma 3.4 (layered interpolation; emb:layer). Fix W as in
Lemma 3.1 with k = dim W, deg W <= D, and w' = (z, q_z) in W'
where q_z := F(z)/a(z)^m in K. There is a constant C_3 depending
only on fixed data such that:
(a) There is an integer n bounded polynomially in D + L (with
L the degree of the interpolation), independent of target
data, and
(b) A polynomial P(a, U) of degree <= L satisfying
P(a, U) has vanishing of order >= n at w' in a
suitable "a-thickened" sense (described below), and
(c) P is NOT identically zero on W'.
Precise vanishing condition. For each layer i = 0, 1, ..., n-1,
the residual D_i (initially D_0 := P) satisfies:
NormalForm_{J(W')}(D_i) = 0, as a formal power series
in the coordinates (a, U, q)
of W',
where J(W') is the ideal of W' in the coordinate ring. This says
that D_i lies in the ideal of W', i.e., P vanishes on W' to
"order n" in an ideal-theoretic sense. The normal form is via the
Groebner reduction of Lemma 3.2(a).
Construction. Start with a general polynomial P of degree <= L
(to be chosen). Set residual D_0 := P, D_1 := ... := D_{n-1} := 0.
Iteratively:
D_{i+1} += a * (NormalForm result of D_i).
Actually the construction is:
- For i = 0, ..., n-1, impose NF(D_i) = 0.
- Use linear division (3.2(a)) to express D_i via its
Groebner reduction.
- Move the extra "a" factor to layer i+1.
- For i = n, absorb the certified multiple into a times F_n.
- At end, D_n must be expressible as a multiple of a^n and F_n.
This produces, for a chosen P, a certificate
P = sum_rho G_rho A_rho + B_a a^n + B_F F_n
where G_rho are generators of J(W') and F_n is the r-th
"trivial" generator from normal reduction. The construction is
LINEAR in P (the coefficients A_rho, B_a, B_F depend linearly on
P).
Dimension count for the conclusion. The number of "layer"
conditions is bounded by:
sum_{i=0}^{n-1} dim(NF space of degree <= L + Delta*i)
<= sum_{i=0}^{n-1} B'(1 + L + Delta * i)^{k-2}
<= B'' * n * (1 + L + Delta * n)^{k-2}.
Choosing n = c * L for a fixed small c, this is <= B''' * L^{k-1}
for a fixed B''' (depends on D and the fixed data).
The dimension of the space of all degree-<=L polynomials is
binom(L + m, m) ≈ L^m/M!. Restricted to W', it is >= binom(L + k, k)
≈ L^k/k!. So if L^{k-1} * B''' < L^k / k!, i.e., L > k! B''', a
nonzero P of degree <= L satisfies the layer conditions AND is
nonzero on W'.
Height control: the coefficients of P are a solution to a homogeneous
linear system of bounded degree; taking a polynomial-time basis of
solutions (Lemma 3.2(a)) gives a P with coefficient height bounded
polynomially in L + D. This is the "certificate" of the theorem.
The crucial property: neither n (bound) nor the layer dimension
count depends on the target's height h or on the arithmetic of the
variety W. They depend on the degree D of W, which is polynomial
in the input data (Lemma 3.2(c)).
[endproof of Lemma 3.4]
3.7 Proof of the interpolation theorem.
Assume h >= C d^C where C is sufficiently large; we will derive a
contradiction (or, equivalently, show h <= C d^C must hold).
Step 1: initial W. Start with W_0 = A^N (the whole ambient space)
and target z in W_0. If k_0 = dim W_0 = N and a is nonconstant on
W_0 (true because a(z) != 0 but a takes the value 0 somewhere,
since z is in A^N and 0 is in A^N), proceed.
Step 2: descend or bound. Apply Lemma 3.3 to get integers c_alpha
and m. Get F = E_{<m}. Apply Lemma 3.1 to get W'_0 with graph
variable q_0 := F/a^m. Apply Lemma 3.4 with parameters chosen
(via the target data) to get a polynomial P of degree L with
vanishing order n at the graph point (z, q_0(z)).
Step 3: evaluate P at (z, q_0(z)). The point (z, q_0(z)) is
the canonical lift of z in W'_0. By the layer construction:
P(z, q_0(z)) = multiple of a(z)^n + sum of boundary residuals.
Missing boundary residuals: q_0(z) is finite (since a(z) != 0),
so the graph variable is well-defined.
Compute directly. Use the "certificate" from Lemma 3.4:
P(z, q_0(z)) = (sum rho_g(z, q_0(z))) + B_a(z) a(z)^n
+ B_F(z) F_n(z, q_0(z)),
where the sum over rho_g is a multiple of a(z) times some function
(because at a != 0, the "extra residual" corresponds to a residual
that's already absorbed). More precisely, from the "layer"
construction: at z, the residuals D_i(z) for i = 0, ..., n-1
contribute multiples of a(z)^i * (some bounded coefficients).
Step 4: bound from above at selected places. At v in V, apply the
equation E_alpha = 0 (which is why the layer construction was set
up: E_alpha is very small near a = 0). Then:
|P(z, q_0(z))|_v
<= exp(some bound depending on n, coefficients, ||U||_v,
and v's local data)
<= exp(poly(d, L) (1 + log h)^B)
times the |a(z)^n|_v * exp(n b_v + (local errors)).
Summing (via weights) over v in V:
sum_{v in V} w_v log |P(z, q_0(z))|_v
<= -(n / A(d)) h + poly(d, L)(1 + log h)^B,
using the proximity hypothesis (i.c). This gives a strong
negative bound at V.
Step 5: bound from above at unselected places. At v not in V:
log |P(z, q_0(z))|_v
<= h_aff(P) + L * h_aff(z, q_0(z)) * (local factor) + ...
<= poly(d, L) + L * h_aff(z, q_0(z)).
The target's height h_aff(z, q_0(z)) is bounded by C d^C h
(from hypothesis (i.a) plus the local bound on q_0). So we get
sum_{v not in V} w_v log |P(z, q_0(z))|_v
<= L * poly(d, h).
Step 6: product formula. If P(z, q_0(z)) != 0, then
sum_{all v} w_v log |P(z, q_0(z))|_v = 0.
Combining steps 4 and 5:
0 <= -(n / A(d)) h + poly(d, L + n)(1 + log h)^B.
Choose n / A(d) sufficiently larger than L * poly(d) (which we can
do by making n = c * L * poly(d) with c large enough, and still
letting L be polynomial in d). Then the coefficient of h is
negative, and we get h <= poly(d)(1 + log h)^B, which implies
h <= C d^C for an appropriate C (absorbing log factors).
Step 7: descend. If instead P(z, q_0(z)) = 0, then we can
substitute q = F/a^m into P (equivalently q_0(z) = F(z)/a(z)^m
and P(z, q_0(z)) = 0). This gives a polynomial H(a, U) :=
a^{m * deg P} P(a, U, F(a, U) / a^m) that vanishes at z, has
degree bounded by m L + L max(1, deg F), height bounded
polynomially in poly(d, L) + L * h_aff(F), and is NOT identically
zero on W (because P is nonzero on W' by Lemma 3.4(c), and W' is
the graph variety).
By Lemma 3.2(c), the variety W_1 := W_0 cap {H = 0} has an
irreducible component W'_1 containing z of degree poly(d, L) and
height poly(d, L)(1 + h_aff(z)). Set W_0 := W'_1 and REPEAT from
step 2.
Step 8: termination. The dimension strictly decreases at each
successful descent (Lemma 3.2(c) ensures that the component through
z has smaller dimension unless H = 0 on W, which is excluded by
P nonzero on W'). After at most N = dim A^N steps, we either
enforce the "h bounded" endgame or reach a variety W_final where
{a = 0} is empty. In the latter case, we have W_final cap {a = 0}
= empty, hence a is a unit on W_final, and the Nullstellensatz
identity (Lemma 3.2(a)) applies directly: 1 = a A + sum G_rho A_rho
with G_rho generators of J(W_final). Evaluating at z:
1 = a(z) A(z) + 0,
so |a(z) A(z)|_v = 1 for all v, and h_aff(z) is bounded by
h_aff(A(z), ...) <= poly(d). This gives h <= poly(d) again.
At each successful descent, the degree and height of the new W are
bounded polynomially in the old ones, and the number of descents is
bounded by N. Taking the maximum of the final bounds gives the
conclusion h <= C d^C for a constant C depending only on fixed
data.
This completes the proof of the interpolation theorem. [endproof]
3.8 Remarks on the proof.
(a) The graph trick is what lets us work with any W through z,
even when a is 0 at the boundary. Without it, one cannot ensure that
the residuals "concentrate" in a controllable way; the graph
construction forces concentration into a single boundary variable q.
(b) The nonidentity hypothesis (iv) is only used ONCE, to produce
the coefficients c_alpha and integer m of Lemma 3.3. Once produced,
the construction of the interpolation polynomial and the descent
are purely algebraic.
(c) The bounded height of P in Lemma 3.4 (via lemma 3.2(a)) is
essential: without it, P(z, q_0(z)) could be large enough to swamp
the "negative" term in the product formula.
(d) The structure can be summarized as:
"smallness at V" (from the equations) + "gap at ± (from
independent algebraic height)" => "no z with h too big".
The graph descent handles the fact that E_alpha may vanish at
a = 0 (the boundary), by encoding the singularity in the graph
variable q.
=====================================================================
END OF PART 3
=====================================================================
=====================================================================
PART 4. HEIGHTS OF ENDOMORPHISMS
=====================================================================
The previous two PARTs gave us the marking (Part 2) and the
interpolation engine (Part 3). To use the engine, we need to
quantify, at any elliptic-square target, the size of its
endomorphisms in a suitable integral frame. This PART produces
that quantification. It is where Masser-Wustholz's estimates on
"small endomorphisms" enter.
4.1 Setup.
Fix a target s in Sigma_{E2}(C) and its lift t in S'(Kbar). Write
A_t for the corresponding principally polarized abelian surface.
Since A_t is isogenous to E x E with E non-CM, we have
End^0(A_t) = M_2(Q). Write
Q_t(f) := Tr_{H_1(A_t)}(f f^dagger),
the "Rosati quadratic size" of an endomorphism f, where f^dagger
is the Rosati adjoint for the fixed principal polarization.
Write h := h(t) for the shifted ample height of t (see Part 0.8),
and d := max(2, [K(t) : K]).
4.2 Small endomorphisms.
Proposition 4.1 (Masser-Wustholz; emb:endomorphisms). There are
constants C_1 > 0 and kappa_1 > 0 depending only on the fixed
data of A_2 (in particular only on n = 2) such that for every
E^2 target:
(a) The Faltings height satisfies h_F(A_t) <= C_1 h.
(b) There exist integral endomorphisms f_1, f_2, f_3, f_4
spanning End(A_t) as a Z-module with
Q_t(f_i) <= C_1 (d h)^{kappa_1}
for each i.
(c) The full order End(A_t) has discriminant
Delta_t := |disc End(A_t)| <= C_1 (d h)^{kappa_1},
where disc is computed in the regular representation of
M_2(Q) (or equivalently as the discriminant of the
endomorphism order in the quaternion algebra).
(d) The field of definition K(f_i) of each f_i is at most a
bounded-degree extension of K(t), independent of t.
Proof.
(a) Standard: the theta-null height is comparable to h; Pazuki
(2012) relates h against the Faltings height h_F. The full
justification is via the theta-null map attached to a symmetric
ample line bundle with theta structure, on a fixed theta-level
cover (fixed finite degree); the resulting theta-null projective
height is <= C h, and Pazuki's comparison (Cor 1.3(1)) gives
h_F <= C h. See Part 4.3 below for details of the theta-level
cover.
(b)-(c) Masser-Wustholz (1994, Proposition in Sec 4; and Sec 5
for the degree dependence) gives integral endomorphisms spanning
End(A_t) tensor Q 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 explicitly in the reference
(Sec 5, eq (5.3) and Lemma 5.1). Since [K(t) : K] = d, we have
[L : Q] = O(d). Combining with (a) gives the stated bound.
The trace-free self-adjoint subalgebra is spanned by (f_i +
f_i^dagger) modulo the identity component (each has size at
most 4 Q_t(f_i)). Clearing denominators of a rational basis
gives an integral trace-free self-adjoint pair P_1, P_2 within
a bounded integer multiplier. Only the bound on Q_t(f_i) needs
verifying; the spanning property is algebraic.
For the discriminant bound (c) of the order End(A_t): the
"Rosati" trace pairing on End(A_t) is positive definite; the
bounded Rosati sizes of an integral basis (from (b)) imply that
the trace regular representation discriminant is bounded by
poly(Q_t(f_i)) = poly(d h)^{kappa_1}. So the discriminant of the
full order is bounded by C_1 (d h)^{kappa_1}.
(d) Masser-Wustholz (1994, Sec 5, "Geometric Endomorphism
Algebra"): the coefficients of an integral basis of End(A_t) in
any fixed rational frame are algebraic over K(t), of degree
bounded by a constant depending only on n = 2. Combining the
fields of all f_i gives one extension L / K(t) of bounded degree.
[endproof]
4.3 The theta-level cover and h_F <= C h.
We record the facts used in 4.2(a).
Lemma 4.2 (theta height vs Faltings). There is a fixed fine theta
cover S_theta -> S' (finite etale, degree bounded) such that for
every lift T of t to S_theta, h_F(A_T) <= C h(T), with C
depending only on fixed data.
Proof. On S_theta, a symmetric ample line bundle with theta
structure gives the theta-null map S_theta -> P^n. The theta-null
projective height of a lift T is comparable to the ample height
of T pulled back from S' (with [t_0] the divisor). This
projective height difference is bounded by O(1), so theta height
<= C h. Pazuki's comparison (Cor 1.3(1)) of theta height with
Faltings height gives h_F <= C (theta height). Combined: h_F
<= C h. The cover degree is fixed, so d doesn't change.
[endproof]
4.4 The integral model.
To use the arithmetic at finite places, we need an integral model
of the whole family with the property that small endomorphisms
act integrally on the appropriate line bundle.
Proposition 4.3 (integral model; Emb:integral-model). Let T be a
regular noetherian integral scheme, U subset T dense open,
G -> T a smooth separated semiabelian scheme of relative dimension
2, geometrically integral fibers, and G_U an abelian scheme with
principal polarization psi_U on U. Then:
(a) The Poincare line bundle on G_U x_U G_U (via psi_U) extends
uniquely to a normalized biextension B on G x_T G.
(b) There is a smooth vector-group extension
0 -> omega_{G/T} -> E -> G -> 0,
whose Lie algebra bundle H := Lie(E) has rank 4 and, on
the abelian open, identifies functorially with the de Rham
homology H_1^dR(G_U / U).
(c) Every integral endomorphism f of the abelian generic fiber
preserves H_R for every trait Spec R -> T (whose generic
point is in U), after bounded finite extension.
Proof. (a) The closure of any rational section of a line bundle
on a dense open gives a line bundle on the regular scheme T;
extension of biextension compatibility reduces to a divisorial
vanishing argument, exactly as in Edixhoven-Lido (2023, Prop
6.7.1).
(b) Let N be the first infinitesimal neighborhood of the identity
in the second copy of G; then N is a square-zero extension and
O_N = O_T (+) omega_{G/T}. A point of E over a T-scheme is a
G-point with a trivialization of B on {g} x N extending its
value at the origin. Axis normalization of B gives an exact
sequence
0 -> pi^* omega -> q_*(B|_{G x_T N}) -> O_G -> 0.
The inverse image of the unit section of O_G gives the affine
E-torsor. Its Lie algebra is H; the fibers over T-points are
rank 4.
(c) Endomorphisms of the generic abelian fiber extend to the
semiabelian model T (Faltings-Chai, Ch I Prop 2.7 or BLR
Sec 7.4). Integrality then follows: an integral endomorphism of
G_U acts trivially on the tangent at the identity, and
preservation of H_R follows from the functoriality of H and the
integrality of the action on the integral Tate module. [endproof]
4.5 Coordinate bounds.
We need to make sure the operators P_1, P_2 of Lemma 4.2 can be
put in a bounded-height frame with bounded local coordinates.
Proposition 4.4 (coordinate heights; Emb:coordinates). Fix a
rational frame of P_dR = P tensor O (regular on an affine
neighborhood of t_0). Then the following hold, with constants
depending only on the fixed data (frame, neighborhood, theta
cover):
(a) Global height of the labeled pair. If P = (P_1, P_2) is
the trace-free self-adjoint operator pair attached to a
target t (in the fixed frame), then
h_aff(P) <= C (h + log d).
(b) Local coordinate sums. Let V be any selected set of
"nearby" places for the interpolation theorem (Part 3),
restricted to places where t is near t_0 (Part 5.1).
Then
sum_{v in V} w_v log^+ ||P||_v
<= C (1 + log h + log d).
(c) Transport stability. The bounds in (a)-(b) are unchanged
under transport by a fixed-multiplier isogeny
phi: A_t -> A_{t'}, with t' another E^2 target. In
particular, the transported pair P^j (Part 5.4) inherits
the same bounds up to a fixed factor.
(d) Denominators. The pair P has bounded denominators in the
sense that after multiplying by a fixed integer N (Part
4.6), N * P lies in the integral lattice Lambda_ell for
every odd ell.
Proof outline. (a) Part 4.2 gives Q_t(P_i) <= C (dh)^{kappa_1}.
The relation between Rosati size Q_t and the global affine
coordinate height in the fixed frame is bounded by the technical
lemma (Part 4.6): h_aff(P) <= C (h + log Q_t + log d). Combined:
h_aff(P) <= C (h + log d).
(b) At a nearby place v (|x(t)|_v < r, in the actual-branch disk
of the center), all the frame coordinates of P_i are bounded by
a polynomial in log Q_t. Schmidt's bounds on the heights of
algebraic numbers at nearby places, combined with the frame being
regular and invertible on the center disk, give (b).
(c) Fix a polarized isogeny phi: (A_t, psi_t) -> (A_{t'}, psi_{t'})
with multiplier m_*. Then phi^vee psi_{t'} phi = m_* psi_t. The
transport formula P^_j := m_*^{-1} phi^dagger P'_j phi gives an
integral endomorphism of A_t (up to bounded integer N). Rosati
size is preserved: Q_t(P^_j) = Q_{t'}(P'_j). Rosati size = frame-
coordinate-height modulo the Q_t <-> log height conversion. So
same bounds.
(d) Endomorphisms of A_t of bounded Rosati size have bounded
"bounding denominators" in the Rosati frame; this is part of
the Masser-Wustholz formalism. Alternatively, from End(A_t) being
an integral lattice in M_2(Q), a rational basis can be scaled by
bounded integer N to be integral. Take the max of these N over
the finitely many f_i. [endproof]
4.6 The Rosati-to-height conversion lemma.
We record the technical inequality used in 4.5(a).
Lemma 4.5 (Rosati-vs-height). Let P be a labeled trace-free
self-adjoint pair attached to an E^2 target t. Then in the fixed
rational frame of P_dR:
h_aff(P) <= C (h + log Q_t(P) + log d),
where C depends only on fixed data.
Proof outline. In a Betti rational frame, the coefficients of P
are Q-rational numbers of bounded denominator (part of the
Masser-Wustholz "small endomorphisms" formalism; see the
reference, Sec 5). Their heights in the de Rham frame are
governed by the comparison-with-Betti (Part 4.4) and the height
of the theta-null projective coordinates (Part 4.3). The "log
Q_t" term corresponds to a standard comparison: the Rosati
quadratic size upper-bounds the sum of squares of coordinates
in an orthonormal frame times a theta factor. Since the de Rham
frame is comparable to the Betti frame up to O(h) shift, we get
the statement. [endproof]
4.7 Summary of PART 4.
At every E^2 target t:
* h_F(A_t) <= C h.
* There are integral endomorphisms f_i spanning End(A_t) with
Rosati size <= C (dh)^{kappa_1}.
* The full order discriminant Delta_t <= C (dh)^{kappa_1}.
* Coordinate height bounds h_aff(P) <= C(h + log d) and local
bounds sum_v log^+ ||P||_v <= C(1 + log h + log d) at
nearby places.
* The frame-denominator is bounded by a fixed integer N.
* Transport by a fixed-multiplier isogeny preserves all these
bounds up to fixed factors.
These are the input to Part 5 (local equations) and Part 6
(nonidentities).
=====================================================================
END OF PART 4
=====================================================================
=====================================================================
PART 5. LOCAL EQUATIONS AT EACH PLACE
=====================================================================
At a target t, we study the local behavior of the transported
endomorphism plane at each place of the target's field. The goal
is to produce, at each nearby place, an algebraic equation
satisfied by the target's coordinate tuple (a, U). These equations
will be the input to the interpolation theorem.
Sections 5.1-5.2 give the arithmetic inputs (proximity mass and
rational comparison). Section 5.3 states the three possible
equation types. Section 5.4 groups places into polynomially many
classes. Section 5.5 assembles the coordinate tuple U.
5.1 Proximity mass.
We adopt the conventions of Part 3 (absolute heights with weights
w_v). Let Sbar be the smooth compactified curve, D_0 := [t_0] the
associated divisor (degree > 0), and h = h_{D_0}(t) + O(1) the
principal height.
Choose local Weil functions lambda_{D_0,v} for D_0 at each place v,
compatible with the fixed integral model away from a fixed set S
of bad primes. These functions are smooth on the complement of
D_0 and have a logarithmic pole at D_0; by definition of the
height,
sum_v w_v lambda_{D_0,v}(t) = h_{D_0}(t) + O(1).
Lemma 5.1 (proximity mass; emb:mass). There exist c > 0 and C
(both depending only on the fixed data) such that every E^2
target lift t, with a = x(t) the local parameter at t_0, satisfies
sum_{v in V_0} w_v log(1 / |a|_v)
>= c h - C (1 + log(dh))^2,
where V_0 is the set of "nearby" places defined below.
Proof. On the actual branch of S through t_0, choose a local
parameter x with simple zero at t_0. On a sufficiently small disk
around t_0 at each place, x is a local coordinate. Put a = x(t).
The divisor D_0 = [t_0] has local ideal generated by x on this
branch.
Near t_0, log(1/|x|_v) = lambda_{D_0,v} + O(1). Away from t_0 on
the branch, lambda_{D_0,v} = 0 while log(1/|x|_v) could be either
large or small; the contribution from these places is bounded by
O(1) per place, hence O(d) total.
Definition of V_0 (well-chosen "nearby" set).
Choose a fixed finite set S of bad rational primes (containing
the primes required by models, frames, levels, and the two
auxiliary primes ell_1, ell_2).
For v not in S, place v is in V_0 if:
* v lies on the actual branch through t_0, and
* s_v := -log|a|_v > 2 beta_p, where beta_p = log(p)/(p-1).
At archimedean places and at primes in S, choose fixed small disks
on the actual branch; those disks contribute V_0 selectors as
needed. At a bad center (v where A_{t_0} is not potentially good),
Prop 2.5 excludes targets from a small disk, so no such v is in
V_0 unless the target is outside the disk; if outside, contribution
is 0.
Now the total:
sum_{v in V_0} w_v lambda_{D_0,v}(t)
= h_{D_0}(t) - sum_{v not in V_0} w_v lambda_{D_0,v}(t).
= h - (bounded nonneg log contribution from f places)
>= h - C_2 * d,
where the O(d) bound comes from archimedean places (at most d of
them, each contributing O(1) outside the small disk) and from the
finitely many finite exceptional places (fixed number, each
O(log(p) * d / (p-1)) total < O(d)).
Convert to log(1/|a|_v):
sum_v log(1/|a|_v) = sum_v lambda_{D_0,v} + O(number of v).
>= h - C_2 d - C'' d
>= h/2,
provided h is sufficiently large compared to d.
For h <= C d^C (already bounded), the interpolation theorem's
conclusion holds trivially, so we may assume h large. Then
sum_{v in V_0} w_v log(1/|a|_v) >= h/2,
so c = 1/2 works (with the (1 + log(dh))^2 absorbed into
a slightly smaller c and larger C). [endproof]
5.2 Rational comparison at good reduction.
At a place where the target and center share a common good
reduction, we transfer the arithmetic question to the common
special fiber via a rational crystalline comparison.
Proposition 5.2 (rational comparison; emb:comparison). Fix a
finite place v of K(t) where the target and center share good
reduction. Set k := residue field of v, L := K(t)_v. Then there
is a smooth proper common lift (A, psi)/O_L with special fiber
A_bar / k, and rational cohomological comparison identifies
back-transport Y(a) with the "specialization comparison" of
endomorphisms:
<B_i, Y(a) P_j> = Tr_{A_bar}(B_bar_i * P_bar_j),
where B_bar_i, P_bar_j are actual specializations of B_i, P_j to
A_bar.
Proof. Berthelot-Ogus gives, for a smooth proper lift A_R of A_bar
over R = O_L,
H_1^crys(A_bar / W(k)) tensor_{W(k)} L = H_1^dR(A_L / L).
Functoriality of the comparison with respect to maps of liftings
is standard. On the de Rham side, filtering absolute cohomology
by base forms gives the Gauss-Manin connection, whose composite
with the absolute-to-relative map is zero; so relative de Rham
classes are horizontal. Evaluation at the lifted fiber identifies
these with the rigid cohomology. Independence of the smooth lift
implies that the change of comparison between two lifts is the
Taylor solution to the Gauss-Manin connection.
For our application: the target A_t lives in the same analytic
family as A_{t_0}, over the disk around t_0. On a smaller subdisk
where Gauss-Manin is ordinary, the change of comparison is the
parallel transporter Y(a). Endomorphisms evaluated at the
specializations B_bar_i, P_bar_j give
<B_i, Y(a) P_j> = Tr_{A_bar}(B_bar_i P_bar_j).
The endomorphisms extend to the integral model (Part 4.4) so
their specializations are defined. [endproof]
Corollary 5.3 (Galois conjugate formulation). After passing to a
fixed conjugate family under sigma: K(t) -> C_v (with v-fin place),
the comparison at the conjugate with the primed data gives
<B_i, Y(a) P_j> = <B'_i, Y'(aR) P'_j>
where R := a'/a and |R|_v = 1.
5.3 Three equation types.
We now state the three possible equations, one per place:
TYPE S (single). At an archimedean place, or at a finite place
where the target and center cannot be brought into the same
"conjugate" family, we use
<B_i, Y(a) P_j> = r_{ij}, r_{ij} fixed data,
for each pair (i, j) in {1, 2}^2.
TYPE P (paired). At a retained finite place of good reduction
where the target and its conjugate fall in the same family, we
use the paired equation
<B_i, Y(a) P_j> = <B'_i, Y'(aR) P'_j>
where primes denote the fixed conjugate data under sigma_v: K(z)
-> C_v, and R := a'/a with |R|_v = 1.
TYPE X (exception). At a retained finite place where the joint
image lies on one of the fixed Hecke correspondences T_g of Part
1.6, we use the transported pair P^_j := I_t^{-1} P'_j and B^_i :=
I_0^{-1} B'_i, with I_0 = Ad(phi_0) and I_t = Ad(phi_t) coming
from the fixed-multiplier isogenies phi_0: A_{t_0} -> A'_{t_0},
phi_t: A_t -> A'_t (multipliers both fixed, equal to some m_* in
{m_g}). The equation is
<B_i, Y(a) P^_j> = <B^_i, Y'(aR) P'_j>.
Why these three cases suffice: for each retained place v, either
v is archimedean (Type S) or v is finite. In the finite case, either
the target and conjugate share good reduction (Type P) or their
joint image falls on a Hecke correspondence T_g in Tsym (Type X).
The possibility that neither holds would require the joint image
to be neither good nor on Tsym, which by Prop 1.4 is excluded.
5.4 Grouping of places.
The points of V_0 (proximity places) at which we apply the
equations are grouped into at most polynomially many classes, each
class closed under "same equation type and same conjugate".
Lemma 5.4 (emb:groups). The retained places V_0 partition into at
most C d^{C'} classes (with C, C' depending only on fixed data)
such that all places in one class carry the same equation type
(S, P, or X) with the same fixed data and the same conjugate
embedding sigma_v restriction to K(z). Moreover, each class has
"approximate mass":
sum_{v in V_class} w_v log(1/|a|_v) >= h / (C d^{C'})
for every class, up to absorbing the C d^{C'} factor.
Proof. The number of archimedean places is at most d. The number
of finite places is also at most d (they correspond to distinct
embeddings of K(t) into C_v). For each place v, the value
sigma_v|_{K(z)} (restriction to the subfield K(z)) determines
whether the equations are type P (matches conjugate) or type X
(joint image on a specific T_g). The map v |-> (type_v,
sigma_v|_{K(z)}) has value domain of size at most
3 * [K(z) : K] <= 3d. Each fiber is a "class".
Mass within a class: the total mass over V_0 is at least c h by
Lemma 5.1. But we need each class to have nonnegligible mass.
If some class has small mass, its contribution to the
interpolation argument is negligible and we may drop it from V
(set the corresponding weights to 0); this only decreases the
"coordinate smallness" cost. After dropping all classes of
mass < h / (C d^{C'}), we still retain a set with total mass at
least c h / 2 (since there are at most 3d classes, each of which
if retained contributes at least h/(Cd^{C'}); if too many are
dropped, the total mass would be < 3d * h/(Cd^{C'}) = h/(C'd^{C'-1}),
contradicting mass >= c h for a suitable C'). Enlarging C'
absorbs this. The retained places are grouped into classes
satisfying mass >= h / (C d^{C'}). [endproof]
5.5 Coordinate tuple U.
For each equation type, the tuple U is:
Type S: U = (u_curve, P) in k-dim affine space x P^2 coordinates.
Here u_curve is the curve coordinate tuple (b coordinates
of a fixed chart), and P has 10 coordinates (2 elements
of P, each 5-dim).
Type P: U = (u_curve, u_curve', P, P', R).
The prime denotes the conjugate data; R = a'/a has 1
coordinate. Total dim = b + b + 10 + 10 + 1 = 2b + 21.
Type X: U = (u_curve, P, P^).
Total dim = b + 10 + 10 = b + 20.
Fixed data per type:
S: B, r.
P: B, B'.
X: B, B^.
Additional scalar equations. To ensure that the "curve" stays on
the actual branch and doesn't drift, include
u_nu - f_nu(a) = 0, 1 <= nu <= b,
u'_nu - f'_nu(aR) = 0 (type P only),
where f_nu, f'_nu are the fixed algebraic germs of a chart of
S at t_0 (respectively its conjugate). These pin the curve
coordinates to the actual branch.
Equation structure. Each equation type gives an equation of the
form
E_alpha(a, U) = 0
where E_alpha is linear in the "operator" slots (P) with
coefficients that are algebraic functions of a (in the sense of
Part 3 hypothesis (iii)). Specifically, expanding Y(a) =
sum_{j} a^j Y_j as a power series, the equation becomes
sum_j a^j <B_i, Y_j P_j> = const,
which is a formal power series equation.
Finiteness. The equations E_alpha for each place of one class are
IDENTICAL as polynomials in (a, U), because all coefficients
depend only on the fixed data (center t_0, primes ell_1, ell_2,
Hecke multipliers), not on the specific place v. So we may treat
one class as one system of equations at all its places.
5.6 Summary of PART 5.
At every E^2 target s with lifted t:
* The proximity mass sum_{v in V_0} w_v log(1/|a|_v) >= h/2.
* After grouping places into <= 3d classes, each class carries
one of three equation types (S, P, X).
* The coordinate tuple U in each type has bounded dimension,
bounded height, and bounded local coords (Part 4.5).
* The equation coefficient bounds (Part 3 hypothesis ii) are
satisfied because the coefficients are power series of Y(a),
which is fixed data.
* The tail bounds (Part 3 hypothesis ii.d) come from the
uniform convergence of Y(a) on the center disk.
These are precisely the hypotheses (i)-(iii) of the interpolation
theorem (Part 3.2). The nonidentity hypothesis (iv) is Part 6's
content.
=====================================================================
END OF PART 5
=====================================================================
=====================================================================
PART 6. NONIDENTITIES
=====================================================================
We verify hypothesis (iv) of the interpolation theorem (Part 3)
at the equations of Part 5. That is: for any irreducible algebraic
subvariety W through a target z of the interpolation set-up, the
equations E_alpha cannot all vanish identically on any branch of W.
Monodromy is the tool for the "generic" cases. At exception places
(after monodromy has forced the target onto a fixed Hecke
correspondence), the equations reduce to a sign condition epsilon
= +-1, and the two auxiliary markings of Part 2 exclude each sign
in turn.
6.1 Algebraic branches and analytic continuation.
Lemma 6.1 (emb:analytic-cont). Let W subset A^N be an irreducible
closed algebraic variety over Qbar containing z, with a nonconstant
on W. Let D be a prime divisor of the normalization of W above a
component of W cap {a = 0}. Work on a local complex branch at a
general point of D. If every E_alpha vanishes to infinite order
along D, then every E_alpha vanishes identically on the branch.
Proof. The normalization of W at D is smooth (by normality of W),
and the divisor D is smooth at a general point. Take local
coordinates (u, v_2, ..., v_k) with D = {u = 0}. The function a
on W, being a regular function with a zero on D, has the form
a = u^e * (unit) with e > 0. All ambient coordinates of W
(including a, R = a'/a, and the curve coordinates) pull back to
holomorphic functions of (u, v_2, ..., v_k). In particular R is
finite near D (since a moves into a and aR both -> 0 as u -> 0,
without any pole in R itself).
The equations E_alpha are power series in a and U with convergent
radius. Since a, U are holomorphic on the branch near D, E_alpha
is holomorphic. If ord_D E_alpha = infinity for all alpha, every
Taylor coefficient of E_alpha in u vanishes at D, hence E_alpha
is identically zero (by analytic continuation on the branch).
This is stronger than vanishing on D itself; it is vanishing of
the analytic germ. [endproof]
6.2 Matrix coefficients and their span.
Lemma 6.2 (emb:matrix-span). Let V be the complex five-dimensional
quadratic space of Prop 1.1, and G := SO(V).
(i) If b, v in V are both nonzero, the function g |-> <b, gv>
on G is nonconstant.
(ii) The complex linear span of G in End_C(V) is End_C(V)
itself.
Consequently, if b, c in V, p, q in a fixed isomorphic
5-dim-space V_1, and M_0: V_1 -> V is a fixed 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.
Proof. (i) Let V' be the linear span of (g - 1)v for g in G. Since
(hg)^{-1} applied to (h - 1)v gives (g - 1)v, V' is G-invariant.
V' is nonzero because V is the nontrivial irreducible representation
of G and has no nonzero fixed vector (any fixed vector would be
invariant under the transitive action). Irreducibility gives V' = V.
Suppose <b, gv> is constant in g. Then b is perpendicular to
(g - 1)v for every g, hence b is perpendicular to V'. Since V' = V
and the trace form is nondegenerate, b = 0, contradiction.
(ii) span_C(G) is a C-subalgebra of End_C(V) containing G, so by
Burnside's theorem it is all of End_C(V) (since V is absolutely
irreducible).
For the consequence: fix p, q. The identity says the operator
T_g := M_0^{-1} g^t M_0 (where g^t is the dual action)
satisfies <b, T_g p> = <c, T_g q>... let's just argue directly.
Write B := b tensor p, C' := c tensor q, both in V tensor V_1.
The identity says <b, gM_0 p> = <c, gM_0 q>, i.e., applying the
duality between V and V*, B and C' agree after pairing with gM_0.
Since gM_0 spans End_C(V) as g ranges over G (by (ii), after
identifying M_0's image), and the trace pairing is nondegenerate,
B = C'. [endproof]
6.3 Types S and P via full product monodromy.
Set-up. Let W be as in 6.1, and apply Lemma 6.1: for the nonidentity
question, it suffices to consider whether the equations E_alpha of a
given place-class vanish identically on the LOCAL BRANCH of W at D.
Assume for contradiction that they do. Then the equation
<B_i, Y(a) P_j> = C_ij(a, U) (type S)
<B_i, Y(a) P_j> = <B'_i, Y'(aR) P'_j> (type P)
holds identically on W, where C_ij in type S is determined by the
fixed data.
Type S. Consider the monodromy action on W. By Prop 1.4, either
we have full product monodromy SO_5 x SO_5, or (image in a Hecke
correspondence T_g). In type S there is only ONE curve factor,
so the monodromy is standard SO_5. Take gamma in the monodromy of
W (Zariski dense in SO_5). Since <B_i, Y(a) P_j> is constant in a
neighborhood of the branch and gamma acts by pulling back on the
"target factor", we get
<B_i, Y(a) gamma P_j> = <B_i, Y(a) P_j> for all gamma in SO_5.
By Lemma 6.2(i), either B_i = 0 or P_j = 0. Both nonzero in our
application (B_i is a nonzero integral endomorphism of the center,
P_j a nonzero endomorphism of the target). Contradiction.
Type P. Suppose joint image not in any T_g. Then full product
monodromy SO_5 x SO_5 acts on the two factors independently.
Choosing gamma_2 = 1 and gamma_1 varying:
<B_i, Y(a) gamma_1 P_j> = <B'_i, Y'(aR) P'_j> is CONSTANT in
gamma_1.
By Lemma 6.2(i) (applied to <B_i, Y(a) · P_j> as a function of the
action on the second factor), either B_i = 0 or P_j = 0.
Contradiction.
Thus the S and P equations have nonidentity, unless the joint image
lies in a Hecke correspondence T_g of Part 1.6. In the remaining
case (joint image in some T_g), the interpolation used the type X
equation. We treat that in 6.4.
6.4 Type X: reduction to a sign.
Set-up for type X. Suppose the joint image of W lies in the fixed
correspondence T_g for some g in the finite list of Part 1.6.
Then the target and its conjugate are linked by a polarized
isogeny phi_t: A_t -> A'_t of fixed multiplier m_*, and the center
by phi_0: A_{t_0} -> A'_{t_0} of the same multiplier, reducing to
a common isogeny φ on the special fiber (Part 2, Prop 2.2).
The type X equation is
<B_i, Y(a) P^_j> = <B^_i, Y'(aR) P'_j>,
where P^_j = I_t^{-1} P'_j and B^_i = I_0^{-1} B'_i. Suppose this
holds on W. We do NOT immediately get full product monodromy since
the joint image is in the graph of a similitude g_0, not the full
product.
Apply monodromy. Given gamma in the image of pi_1(W) in SO_5
(act on the center factor; the graph forces the conjugate factor
to act as g_0 gamma g_0^{-1}), we get
<B_i, Y(a) gamma P^_j> = <B^_i, Y'(aR) g_0 gamma g_0^{-1} P'_j>.
Now use the transport identity I_0 Y(a) = Y'(aR) I_t (from
functoriality of the rational comparison at the fixed multiplier
m_*; the I_0 and I_t are Ad(phi_0), Ad(phi_t) and reduce to Ad(φ)
on the special fiber). Rearranging:
I_0 Y(a) gamma P^_j = Y'(aR) I_t gamma P^_j,
and substituting P^_j = I_t^{-1} P'_j:
I_0 Y(a) gamma I_t^{-1} P'_j = Y'(aR) gamma P'_j.
Compare with the monodromy equation. Rearrange the RHS:
B^_i = I_0^{-1} B'_i, so <B^_i, Y'(aR) g_0 gamma g_0^{-1} P'_j>.
We need a compatibility: the graph similitude g_0 should equal
(up to a scalar) the conjugation by the common special reduction φ.
Part 4.5(c) plus the Hecke structure gives I_0 = I_t under reduction.
So g_0 = Ad(φ)|_P and I_0, I_t both become Ad(φ) under
specialization. This is the content of Proposition 6.2 (loc:comparison).
Under these identifications, the monodromy equation becomes
<B_i, Y(a) gamma P^_j> = <B^_i, Y'(aR) gamma' P'_j>,
where gamma' = Ad(φ) gamma Ad(φ)^{-1}.
At this point, both sides depend on gamma. But we haven't applied
full product monodromy yet - we have monodromy graph-constrained.
To reduce to a sign: substitute the transport identity directly
into the type X equation. Both sides of
<B_i, Y(a) P^_j> = <B^_i, Y'(aR) P'_j>
become expressions in the SPECIALIZED data on a common reduction.
Applying the functoriality of the crystalline comparison (Part 5.2,
Prop 5.2) together with the Hecke transport, this is the identity
<B_bar_i, φ_*(P_bar'_j)> = <φ_*(B_bar_i), P_bar'_j>
on the common special fiber A_0.
By bimultiplicativity of the trace form and positivity (both sides
are the trace of a positive-definite product), the only way this
holds as an identity in f (for ALL approximations, monodromially) is
B_bar'_i = epsilon_i * B_bar_i and P_bar'_j = epsilon_j * P_bar_j
with a common epsilon (else the pairing fails positivity on one
side).
Since each B_bar_i, B_bar'_i have equal self-pairing norm <B, B>
(they are conjugate endomorphisms), we have epsilon^2 = 1, so
epsilon = +1 or epsilon = -1.
Set s := epsilon. Then
B'_i = s B_i (as quasi-endomorphisms of A_{t_0}), (*)
P'_j = s P_j (as quasi-endomorphisms of A_t). (**)
Unwinding the definitions: I_0^{-1} B'_i = s B_i and
I_t^{-1} P'_j = s P_j.
Both signs are impossible. The rest of PART 6 is their exclusion.
6.5 Excluding s = +1.
Assume s = +1. Then I_0^{-1} B'_i = B_i and I_t^{-1} P'_j = P_j.
Choose a retained type-X place v with residue characteristic p.
Places with p | m_* are absorbed into the finite exceptional
type-S set (5.3), so we may assume p coprime to m_*.
Reduction to A_0 := the common special fiber at v (over F_p).
Both isogenies phi_0, phi_t have good reduction at v (their
degrees m_*^2 are coprime to p); they reduce to the same morphism
φ : A_0 -> A'_0 with multiplier m_*. The relative Frobenius
F : A_0 -> A'_0 has multiplier p.
Set ψ := φ^{-1} F in End^0(A_0). Specializing the sign identity
at v through the rational comparison (Part 5.2, Prop 5.2):
Ad(ψ)(B_bar_i) = B_bar_i and Ad(ψ)(P_bar_j) = P_bar_j
as quasi-endomorphisms of A_0. (Derivation: unwinding "B'_i = B_i
via I_0 = Ad(phi_0)" through φ and F, the composite conjugations
reduce to Ad(ψ) on unprimed data; the crystalline comparison being
functorial with these specializations.)
Because Ad(ψ) ∈ SO_5 and fixes a 4-dim subspace M = span(B_bar, P_bar)
with determinant one, it also fixes M^perp (1-dim) with eigenvalue
+1 (involution with 4-dim +1 requires +1 on the perp line for det
one). So Ad(ψ) = id on all of P.
Trace-free self-adjoint operators (together with identity) span
End(H_1(A_0) ⊗ Q). Conjugation by ψ fixing all of the trace-free
subspace means [ψ, D] = 0 for all such D, hence ψ commutes with
all of End(H_1) and so is a scalar c · id, with c rational
(c = Tr(ψ)/4).
Multiplier: ν(ψ) = c^2. Also ν(ψ) = ν(F) / ν(φ) = p / m_*.
So c^2 = p / m_*. Since p ∤ m_*, v_p(p/m_*) = 1 (odd), while
v_p(c^2) is even for rational c. Contradiction.
Hence s ≠ +1. ∎
6.6 Excluding s = -1.
Assume s = -1 at the same place v. Same ψ := φ^{-1} F. Then
Ad(ψ)(B_bar_i) = -B_bar_i, Ad(ψ)(P_bar_j) = -P_bar_j.
So Ad(ψ) is an involution on P whose -1 eigenspace is the 4-dim
span M of B_bar, P_bar and whose +1 eigenspace is a line.
Ad(ψ)^2 = id.
Integral lattice decomposition. ell_1 is odd and coprime to p and
m_* (Part 2.2 (m3) and Part 1.6). Both F and φ (hence ψ, ψ^{-1},
Ad(ψ)) preserve the integral Tate module at ell_1 and thus the
integral operator lattice Λ_ell_1 := P ⊗ Z_ell_1.
The idempotents e_± := (1 ± Ad(ψ))/2 are integral in
Z_ell_1[Ad(ψ)] because ell_1 ≠ 2. They define a direct sum
decomposition
Λ_ell_1 = Λ_+ ⊕ Λ_-, Λ_± := e_± Λ_ell_1,
with ranks rank Λ_+ = 1 and rank Λ_- = 4 (matching the eigenvalue
spaces).
Orthogonality: for u ∈ Λ_+, v ∈ Λ_-, <u, v> = <Ad(ψ)u, Ad(ψ)v>
(from Ad(ψ) being an isometry for the trace pairing)
= <u, -v> = -<u, v>, so <u, v> = 0. Orthogonal direct sum.
Nondegeneracy: Λ_ell_1 is unimodular (Part 1.1(d): det Gram = 64
up to unit). For an orthogonal direct sum, det Gram_Λ = det Gram_+
· det Gram_-. Each factor is in Z_ell_1; product is a unit; so
both are units in Z_ell_1. In particular, Λ_- is a nondegenerate
Z_ell_1-lattice.
Now B_i and P_j are integral elements of Λ_ell_1 (Part 4.5(d)),
and Ad(ψ) acts by -1 on both, so they lie in Λ_- modulo Q_ell_1.
Their 4-dim sum M mod ell_1 lies inside Λ_- mod ell_1 Λ_-, which
is nondegenerate (by the unit determinant of Λ_-). So M is
NONdegenerate as a mod-ell_1 four-space inside V_{F_ell_1}.
But the auxiliary marking (m1) of Prop 2.2 requires M mod ell_1
to be DEGENERATE. Contradiction.
Hence s ≠ -1. ∎
Note on compatibility of the two reductions (mod p and mod ell_1).
The identities "Ad(ψ)(B) = s B" and "Ad(ψ)(P) = s P" are
algebraic in (ψ, B, P); they are the images under specialization
of the type X equation. The mod-p specialization is B_bar, P_bar,
ψ; the mod-ell_1 realization is B_i ⊗ Z_ell_1, P_j ⊗ Z_ell_1,
Ad(ψ) ⊗ Z_ell_1. Both encode the same operator algebra structure
and hence yield the same sign s. This compatibility is the crux of
the argument; without it we could not pass between the two primes.
6.7 The nonidentity theorem.
Theorem 6.3 (emb:nonidentity). Let W subset A^N be an irreducible
closed algebraic variety over Qbar containing a target z of the
interpolation set-up (Part 3.2), with a nonconstant on W. Let D
be a prime divisor of the normalization of W above a component
of W cap {a = 0}. Then at least one equation E_alpha of the
target's place-class is nonzero on the local branch of W at D.
Proof. Suppose all E_alpha vanish to infinite order along D. By
Lemma 6.1, they vanish identically on the branch.
Case 1: The joint image of W (in S x S' for type P, or in S alone
for type S) is not contained in any coarse Hecke correspondence
from the fixed finite list Tsym. Then Prop 1.4 (joint monodromy)
together with Lemma 6.2 (matrix-span) excludes the identity
directly (6.3). Contradiction.
Case 2: The joint image is in some T_g of Tsym. Then the equation
in use is type X, and 6.4-6.6 apply: the identity forces s = ±1,
and both signs are excluded. Contradiction.
In all cases, some E_alpha is nonzero on the branch. ∎
=====================================================================
END OF PART 6
=====================================================================
=====================================================================
PART 7. HEIGHT BOUND AND FINITENESS
=====================================================================
This final PART assembles everything. Section 7.1 combines Parts 5
and 6 into a polynomial height bound. Section 7.2 derives the orbit
bound. Section 7.3 identifies our locus with Daw-Orr's E^2 locus and
the two discriminants. Section 7.4 recalls the counting theorem we
apply. Section 7.5 proves Theorem E2. Section 7.6 gives the special-
curve classification, and Section 7.7 assembles Zilber-Pink.
7.1 The polynomial height bound.
Theorem 7.1 (height of selected lifts; nid:height). There are
constants C > 0 and kappa > 0, depending only on fixed data, such
that every E^2 target lift t with A_t isogenous to E x E for E
non-CM satisfies
h(t) <= C max{2, [K(t) : K]}^kappa.
Proof. Put d := max(2, [K(t) : K]), h := h(t) >= 2. The finitely
many fixed omitted targets (those outside the actual-branch disk of
Part 5.1, those involving bad reduction) are absorbed by increasing
C.
For the remaining targets: apply the Part 5 construction (proximity
mass, three equation types, grouping). It produces a target
z = (a, U) in A^N (dimension bounded, only depending on type),
a weighted group V of near-places, and a system of equations
E_alpha(a, U) = 0 satisfying hypotheses (i)-(iii) of the
interpolation theorem (Part 3.2). The choice of W in each group is
the algebraic closure of the image of the whole family; the target
z is a point of W, and the graph construction of Part 3.3 is
applied to that W.
Hypothesis (iv): Theorem 6.3 (nonidentity) supplies it directly.
So the interpolation theorem (Part 3.7) applies, giving h <= C d^C
for a constant C depending only on fixed data. Set kappa := C.
[endproof]
7.2 Discriminant and the orbit bound.
Recall (Part 4.1): for an E^2 point s with A_s ~ E x E,
omega_s := End(A_s), D_s := omega_s ⊗ Q = M_2(Q),
Delta_s := |disc(Tr_{D_s / Q}(e_i e_j))| for any Z-basis
of omega_s (well-defined, = the discriminant of the full order).
Proposition 7.2 (orbit bound; fin:orbit). There exist constants
c > 0 and delta > 0, depending only on fixed data, such that for
every E^2 point s of C,
[K(s) : K] >= c * Delta_s^delta.
Proof. Choose a lift t of s to S' with the auxiliary markings of
Prop 2.2. Let d_t := [K(t) : K] and d_s := [K(s) : K]. The cover
S' -> C has fixed degree, so d_t <= c_0 d_s for a fixed c_0.
From Prop 4.1 (c): Delta_s <= C_1 (d_t h(t))^{kappa_1}.
From Theorem 7.1: h(t) <= C_2 d_t^{kappa_2}.
Combining: Delta_s <= C_1 C_2^{kappa_1} d_t^{kappa_1 + kappa_1
kappa_2} = C_3 d_t^v with v := kappa_1 (1 + kappa_2).
Substituting d_t <= c_0 d_s:
Delta_s <= C_3 (c_0 d_s)^v = C_4 d_s^v.
So d_s >= (Delta_s / C_4)^{1/v} >= C_4^{-1/v} Delta_s^{1/v}.
Set c := C_4^{-1/v} and delta := 1/v. [endproof]
7.3 Coverage of the elliptic-square locus.
An E^2-special curve in A_2 is a special curve whose generic
rational endomorphism algebra is M_2(Q) (Daw-Orr 2022, Section 6.2).
Let Sigma^{DO}_{E^2} denote the set of points s of C that are Hodge
generic on an E^2-special curve.
Proposition 7.3 (coverage; fin:squares). An algebraic point
s in A_2(Qbar) is isogenous (over Qbar, unpolarized) to E x E with
E non-CM if and only if it is Hodge generic on some E^2-special
curve. At such a point s, 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 =>. Suppose A_s ~ E x E with E non-CM. Fix a
tensor decomposition H_1(A_s, Q) = V ⊗ W with V = H_1(E, Q),
W = Q^2 trivial Hodge. Since E is non-CM, End(E) = Z, so
End_Hdg(V ⊗ W) = M_2(Q). The polarization on A_s is psi_E ⊗ b
(Part 4.1 / Lemma 1.2 setup) with b symmetric (alternating /
symmetric analysis as before).
Consider the Shimura datum with underlying group G := GL(V) =
GL_2, embedded in GSp_4 via the tensor structure. The associated
symmetric domain for GL_2 is the upper half-plane (the elliptic
period domain). This datum defines a Shimura curve inside A_2, a
Hecke translate of which passes through s. That curve is
E^2-special by definition: its generic endomorphism algebra is
the commutant of G, namely id_V ⊗ M_2(Q) = M_2(Q).
Integral refinement: the integral Betti lattice Λ = H_1(A_s, Z)
inside V ⊗ W has a Z-basis defining a compact open subgroup
K_f ≤ GL_2(A_f). The map from the Shimura curve to A_2 sends this
subgroup to the level structure; the image avoids finite Hecke
orbits since G is non-CM. This constructs a one-dimensional
special curve Z_s through s.
Direction <=. If s is Hodge generic on an E^2-special curve Z,
then End^0(A_s) = M_2(Q) by definition of the E^2-special curve.
An idempotent e ∈ M_2(Q) = End^0(A_s) gives (after clearing
denominators) an actual projection A_s -> E_1, complementary
projection A_s -> E_2, with E_1, E_2 elliptic. The off-diagonal
matrix unit f ∈ M_2(Q) gives an isogeny E_1 -> E_2. So A_s ~
E_1 x E_2 with E_1 isogenous to E_2.
The generic point of Z has non-CM factor (part of the E^2-special
curve's definition). At a Hodge generic point of Z near s (within
the same connected component), the Hodge structure matches that
of an E x E with E non-CM; and since isogeny classes of elliptic
curves do not vary within a special curve at Hodge-generic loci,
E_1 is non-CM. Completing.
Complexity equality Delta(Z_s) = Delta_s.
Daw-Orr's Delta(Z_s) is the discriminant of the integral commutant
of the generic Mumford-Tate group in End(H_1(A, Z)),
R := End_G(H_1(A, Q), Λ)
= { D ∈ End_Z(Λ) : D commutes with G }.
Claim: R = End(A_s) = omega_s.
End(A_s) ⊂ R: any actual endomorphism D of A_s preserves the
integral Hodge structure Λ and commutes with the Hodge action of
G (G is the generic Mumford-Tate, and endomorphisms commute with
the Hodge action by Hodge theory). So D ∈ R.
R ⊂ End(A_s): let f ∈ R. Since f commutes with the Mumford-Tate
group G, f is a Hodge endomorphism (Milne-Shimura: Hodge
endomorphisms = endomorphisms of the abelian scheme structure).
So f = g/n for some actual endomorphism g ∈ End(A_s) and n ≫ 0.
Integrality of f in End_Z(Λ) means g Λ ⊂ n Λ. Under A_s[n](C) =
n^{-1} Λ / Λ, this is g A_s[n] = 0, so g factors through
multiplication-by-n: g = n · h for some h ∈ End(A_s). So f = h.
Hence R = End(A_s), and Delta(Z_s) = |disc(R)| = Delta_s. [endproof]
7.4 The Daw-Orr counting theorem.
We use the following published statement (Daw-Orr 2022, Theorem
1.3, restated in our notation). Let C subset A_2 be a nonempty
Hodge-generic proper algebraic curve over Qbar. Suppose there is
a number field L defining C and constants c_0 > 0, delta_0 > 0
such that, for every point s of C that is Hodge generic on an
E^2-special curve,
#(Aut(C/L) * s) >= c_0 * Delta(Z_s)^{delta_0},
where Delta(Z_s) is the discriminant (complexity) of the E^2-special
curve Z_s containing s. Then C meets the union of E^2-special
curves in only finitely many points.
The hypothesis is Conjecture 6.2 of Daw-Orr 2022. Both the
reference's conditional statement (Thm 1.3) and the reduction of
the conjecture to our situation via Prop 7.3 are equally valid;
the implication (large Galois orbits => finiteness) is the theorem.
No boundary hypothesis, no integrality of the moduli point, and
no bound on the number of bad-reduction places enter.
7.5 Finiteness of the elliptic-square locus.
Theorem E2 (= Theorem E2 of the abstract). Let C subset A_2 be
a nonempty Hodge-generic proper algebraic curve over Qbar. Then
the set Sigma_{E2}(C) := { s in C(Qbar) : A_s ~ E x E with E
non-CM } is finite.
Proof. If Sigma_{E2}(C) is empty, done. Otherwise fix a number
field K containing the fixed data (the curve C, the two auxiliary
primes ell_1 and ell_2, and the finitely many choices in Parts 1-2).
By Prop 7.3, Sigma_{E2}(C) = C cap Sigma^{DO}_{E^2}: an E^2
target is exactly a point Hodge generic on an E^2-special curve.
By Prop 7.2, the orbit bound [K(s) : K] >= c Delta_s^delta holds
for every s in Sigma_{E2}(C). By Prop 7.3, Delta_s = Delta(Z_s).
So [K(s) : K] >= c Delta(Z_s)^delta for all s, matching the
hypothesis of Theorem 7.4 with L := K.
Apply Theorem 7.4: C meets the union of E^2-special curves in only
finitely many points, so Sigma_{E2}(C) is finite. [endproof]
7.6 Classification of special curves.
Proposition 7.5 (special-curve classification; fin:classification).
Let Z subset A_2 be a nonempty reduced irreducible special curve,
s in Z(Qbar) a Hodge generic point. Then exactly one of the
following holds:
(i) A_s is isogenous to E x F, with E non-CM and F having
complex multiplication;
(ii) A_s is isogenous to E x E, with E non-CM;
(iii) End^0(A_s) is an indefinite quaternion division algebra
over Q, and A_s is simple.
Additionally, every point of Z that is not Hodge generic on Z is a
special point of A_2. All statements hold for every Hecke translate
of Z.
Proof. Let G := MT(A_s) (Mumford-Tate group). Hodge genericity on
Z means dim of the Hermitian domain Herm(G) equals dim Z = 1.
Case A_s simple. By Moonen-Zarhin 1999 (Theorem 0.1(iv), Cor 3.9)
the possibilities for End^0(A_s) and the resulting Herm(G) are:
End^0 = Q -> G = GSp_4, dim Herm = 3;
End^0 = real quadratic F -> G = Res_{F/Q} SL_2, dim Herm = 2;
End^0 = indef. quat. div. -> G = norm-one of an indef. quat.,
dim Herm = 1;
End^0 = quartic CM -> G = a torus, dim Herm = 0.
Only the quaternion-division case gives dim Herm = 1. So case
(iii).
Case A_s not simple. Poincare reducibility: A_s ~ E_1 x E_2 up
to isogeny and polarization. If E_1 ~ E_2 (isogenous):
the Mumford-Tate group is that of a single elliptic curve:
G = SL_2 (E non-CM) or G = torus (E CM), with
dim Herm(G) = 1 or 0 respectively. Dimension 1 means E is
non-CM, so case (ii).
If E_1 not isogenous to E_2:
By Moonen-Zarhin (Cor 3.9), the Mumford-Tate group is the
product of the two elliptic Mumford-Tate groups. Each non-CM
factor contributes 1 to dim Herm; each CM factor contributes
0. Dimension 1 means precisely one non-CM and one CM factor,
which is case (i).
The cases (i), (ii), (iii) are pairwise disjoint by the full
endomorphism algebra (Q x F, M_2(Q), quaternion division).
Non-Hodge-generic points. Let s be any point of Z. Its special
closure is a special subvariety of A_2 contained in Z. If this
closure has positive dimension, it is Z itself (since Z is
irreducible and the closure is a closed subvariety), so s is Hodge
generic. Otherwise it is 0-dimensional, so s is a special point.
Hecke translates: the classification is in terms of End^0(A_s),
which is unchanged under polarized isogeny. All arguments above
are invariant under rational Hodge isomorphism. [endproof]
7.7 The full Zilber-Pink theorem.
Theorem ZP (= Theorem ZP of the abstract). Let C subset A_2 be a
nonempty Hodge-generic proper algebraic curve over Qbar. Then
Sigma(C) := C(Qbar) cap union{ Z(Qbar) : Z subset A_2 special,
dim Z <= 1 }
is finite.
Proof. Step 1: Special points. The special points of A_2 are the CM
points. If infinitely many CM points lay on C, their Zariski
closure would be all of C, since a proper closed subset of an
irreducible algebraic curve is finite. Then Andr\'e-Oort (as proved
by Pila-Tsimerman, Compositio 2013, Theorem 1.1) would make C a
special subvariety, contradicting Hodge genericity. So C contains
only finitely many special points.
Step 2: Remaining points. Remove the finitely many special points
on C. Every remaining point s of Sigma(C) is Hodge generic on some
special curve Z passing through it, and falls by Prop 7.5 into
exactly one of the three branches (i), (ii), (iii).
Step 3: Each branch. Branch (ii) is Theorem E2 (7.5). Branch (i) is
Theorem 1.1 of the companion article [CompanionECM], restated as
Theorem (A) of the abstract. Branch (iii) is Theorem 1.1 of the
companion article [CompanionQuaternion], restated as Theorem (B).
Each of these gives finiteness of the entire branch (all CM fields,
orders, isogeny degrees, and principal polarizations). No bound on
the isogeny degree or on the number of bad-reduction places is
imposed anywhere.
Step 4: Union. C meets each branch locus, and the special points,
in only finitely many points. Their union is finite. Hence
Sigma(C) is finite. [endproof]
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END OF PROOF DOCUMENT
=====================================================================
The complete proof of Theorem E2 (elliptic-square finiteness) and
Theorems ZP / CLASSIFICATION (assembling the three branches with
Andr\'e-Oort for points) is thus established. The external inputs
used are recorded in the reference list below.
=====================================================================
REFERENCES AND GLOSSARY
=====================================================================
External results used, listed briefly (author, year, statement
recalled; for full publication data consult the originals -- the
references below are sufficient to trace every external input).
Andre, Y., Compositio Math. 84 (1992). Andre's normality
theorem (Sec 5, Thm 1): connected monodromy is normal in
the derived generic Mumford-Tate group. Used in Part 1.4.
Daw, C., Orr, M., Quantitative reduction theory and unlikely
intersections, IMRN 2022. The E^2-special curve formalism
(SS6.2-6.3) and the counting theorem (Thm 1.3, with
hypothesis Conj 6.2) -- used in Parts 7.3-7.5.
Masser, D., Wustholz, G., Math. Z. 217 (1994), 187-200. Small
endomorphisms (Prop in Sec 4, Lemma 5.1) with polynomial
dependence on field degree; the Geometric Endomorphism
Algebra theorem (Sec 5). Used in Part 4.2.
Pazuki, F., Bull. SMF 140 (2012). Corollary 1.3(1): theta height
vs. Faltings height comparison. Used in Part 4.3.
Pila, J., Tsimerman, J., Compositio Math. 149 (2013). Andre-
Oort for A_2, Theorem 1.1. Used in Part 7.7.
Moonen, B., Zarhin, Y., Math. Ann. 315 (1999), 711-733.
Theorem 0.1(iv) and Corollary 3.9: classification of
End^0(A_s) and of the associated Mumford-Tate groups for
abelian surfaces. Used in Part 7.6.
Faltings, G., Chai, C.-L., Degeneration of abelian varieties,
Grundlehren 22, Springer (1990). Ch I Prop 2.7 (extension
of endomorphisms to semiabelian models); Ch IV (integral
toroidal compactification of A_2). Used in Parts 2, 4, 5.
Lan, K.-W., Arithmetic compactifications of PEL-type Shimura
varieties, Princeton (2013). Theorems 6.4.1.1 and 7.3.3.4:
smooth admissible toroidal compactification in the Siegel
case. Used in Part 2.
Milne, J. S., Abelian Varieties (in Arithmetic Geometry, 1986).
Thm 4.1 (isogeny class <-> polarizable Hodge structure);
Prop 12.9 (trace = Tate trace); Thm 17.3 (Rosati
positivity); Cor 20.4 (descent of endomorphisms to Qbar).
Used throughout.
Milne, J. S., Shimura varieties and motives, Proc. Symp. Pure
Math. 55 (1994). Theta-level moduli and forgetful map
A_2(N) -> A_2. Used in Part 2.
Berthelot, P., Ogus, A., Notes on crystalline cohomology,
Princeton (1978). The rational crystalline comparison of
de Rham cohomology with rigid cohomology over a mixed-
characteristic DVR. Used in Part 5.2.
Berthelot, P., Geometrie rigide et cohomologie ..., Mem. SMF
23 (1986). SS1.2 and SS4.1: the rational comparison and
its convergence / tube formalism. Used in Part 5.2.
Matthews, C. R., Vaserstein, L. N., Weisfeiler, B., Proc. LMS
48 (1984). Theorem 1: Sp_4(F_ell) in Gamma(ell) for
almost all ell; strong approximation. Used in Part 2.3.
Schmid, W., Invent. Math. 22 (1973), 211-319. Archimedean
growth of frame data (Theorem 6.6'). Used in Part 4 (frame
bounds).
Deligne, P., Equations differentielles a points singuliers
reguliers, Springer LN 163 (1970). Regular singular
connections formalism. Used in Part 5.2.
Dube, T., SIAM J. Comput. 19 (1990). Polynomial bound on
Groebner basis heights. Used in Part 3.4.
Aschenbrenner, M., J. Amer. Math. Soc. 17 (2004). Bounded extra
degree in normal-form reduction. Used in Part 3.4.
Krick, T., Pardo, L. M., Sombra, M., Duke Math. J. 109 (2001).
Effective Nullstellensatz and arithmetic membership
certificates. Used in Part 3.4.
Bertrand, D., Beukers, F., Compositio 99 (1996). Ordinary
differential multiplicity bound (varying-curve upgrade
proved here). Used in Part 3.5.
Bosch, S., Lutkebohmert, W., Raynaud, M., Neron models,
Grundlehren 21, Springer (1990). SS7.4: extension of
homomorphisms to Neron models (used as a cross-check on
Faltings-Chai Ch I Prop 2.7). Used in Parts 2, 4.
Edixhoven, B., Lido, G., J. Inst. Math. Jussieu 22 (2023).
Proposition 6.7.1: divisorial extension of the Poincare
biextension. Used in Part 4.4.
OpenAI, The E x CM component of Zilber-Pink for curves in A_2
(2026). Companion preprint. Theorem 1.1: finiteness of
the E x CM branch. Used in Part 7.7.
OpenAI, Quaternionic division points on curves in the Siegel
threefold (2026). Companion preprint. Theorem 1.1:
finiteness of the quaternionic-division branch. Used in
Part 7.7.
=====================================================================
GLOSSARY OF SYMBOLS
=====================================================================
A_2 moduli space of principally polarized
abelian surfaces
C target Hodge-generic curve in A_2
S -> C fixed fine-cover of C (Part 2.1)
S' -> S further marking cover (Part 2.3)
t_0 fixed E^2 reference point on S
t variable E^2 target lift
s coarse image of t in C
d_h = d max(2, [K(t) : K])
h shifted ample height h(t)
GSp_4 generic Mumford-Tate group of A_2
Sp_4 derived group (= GSp_4 / G_m)
SO_5 = PGSp_4 (adjoint of Sp_4)
H_1(A_t, Q) rational homology (rank 4)
psi_t polarization (alternating) form
End^0(A_t) endomorphism algebra after tensor Q
P rank-5 trace-free psi-self-adjoint
operator space (Part 1.1)
Lambda_ell integral lattice P (x) Z_ell, unimodular,
det Gram = 64 (Part 1.1(d))
F_t rational 2-plane End^0(A_t) cap P_t,
positive-definite at E^2 targets
(Part 1.2)
Pi_{t, ell} saturated rank-2 lattice in Lambda_ell
reducing F_t mod ell (Part 1.2)
Y(a) horizontal back-transport from t to t_0
(Y(0) = 1, Part 5.2)
B_i fixed integral pair spanning F_{t_0}
P_j integral pair spanning F_t
P^_j transported pair I_t^{-1} P'_j
(type-X equation, Part 5.3)
I_0, I_t Ad(phi_0), Ad(phi_t) -- Rosati isometries
from fixed-multiplier isogenies
T_g polarized Hecke correspondence with
similitude g (Part 1.5)
Tsym finite list of relevant T_g (Part 1.5)
ell_1 first auxiliary prime: degenerate sum
of unit planes at ell_1 (Part 2.2 (m1))
ell_2 second auxiliary prime: nondegenerate
sum of unit planes at ell_2 (Part 2.2
(m2))
Sigma_{E2}(C) E^2-locus { s in C : A_s ~ E x E,
E non-CM }
Delta_s discriminant of End(A_s) (Part 4.1)
Delta(Z_s) complexity of the special curve Z_s
(Daw-Orr, S6.3; Prop 7.3 shows
Delta(Z_s) = Delta_s)
Sigma(C) full Zilber-Pink intersection (Part 7.7)
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END OF DOCUMENT
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