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THE E x CM COMPONENT OF ZILBER-PINK FOR CURVES IN A_2
A complete, self-contained proof
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CONTENTS
0. How to read this
1. The shape of the problem
2. The two theorems
3. The plan of the proof
4. Conventions: heights, periods, essential periods
5. The toolbox: four classical tools and one CM computation
6. Canonical gluing: the integers n and theta
7. Degeneration at a finite prime: the length is log|j_E|/n
8. The boundary of the moduli space and the archimedean estimate
9. The height bound on a curve
10. An essential period on a threefold: the extra power of n
11. Galois orbits
12. Finiteness on the original curve
13. References
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0. HOW TO READ THIS
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The proof contains exactly one new idea. It is in Section 10: take TWO
copies of the non-CM elliptic curve E. Then a single period vector of a
certain abelian threefold sees both independent periods of E at once, so
no integer linear relation can kill it, and the period theorem pays out
an extra factor n^(4/3). Section 9 has produced a bound growing only
like n. The mismatch 4/3 > 1 is the whole argument.
Everything else is either classical, or careful bookkeeping of two
things: how constants behave when the number field grows, and how the
estimates behave when the point walks to the boundary of the moduli
space.
A reader who wants only the logical skeleton should read the statements
of 2.1, 2.2, 6.1, 7.1, 8.1, 8.3, 9.1, 10.1, and then 11 and 12. Those
statements plus their proofs contain every step that is not cited.
ASCII conventions in this document:
x product of varieties, or multiplication of numbers
-> a morphism
<< >> "at most a constant times", "at least a constant times"
O(1) a quantity bounded by an absolute constant
log^+ t max(0, log t)
|x|_v the absolute value at a place v
#S the number of elements of a finite set S
[k:Q] the degree of the number field k
A_s the abelian surface with moduli point s
hF(A) the stable Faltings height of A
h(j) absolute logarithmic height of a j-invariant
D |disc End(F)| for the CM elliptic curve F in play
d [k:Q] for the number field k currently in play
n the gluing index of Section 6 (the product isogeny has
degree n^2)
cplx(s) max{n^2, D} at a mixed point s
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1. THE SHAPE OF THE PROBLEM
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1.1 ELLIPTIC CURVES
An elliptic curve over C is a torus E = C / L, where L = Z + Z*tau is a
lattice with Im(tau) > 0. The number j(tau) classifies E up to
isomorphism. Two elliptic curves are isogenous if there is a surjective
homomorphism between them with finite kernel; after choosing origins
that says one period lattice sits inside the other with finite index.
1.2 ABELIAN SURFACES
An abelian surface is A = C^2 / L, where L is a lattice of rank 4 in
C^2, together with a positive definite Hermitian form H on C^2 whose
imaginary part takes integer values on L x L. The form is a
polarization; it is a principal polarization when its imaginary part is
unimodular on L. Polarizations matter: the same complex torus with two
different principal polarizations is two different points of the moduli
space. We always remember the polarization.
1.3 THE MODULI SPACE A_2
Let H_2 be the Siegel upper half space of degree 2: the symmetric 2x2
complex matrices Z whose imaginary part Y = Im Z is positive definite.
This is an open subset of C^3, so it has complex dimension 3. The group
Sp_4(Z) acts properly discontinuously, and
A_2 = Sp_4(Z) \ H_2
is the coarse moduli space of principally polarized abelian surfaces. It
is a complex variety of dimension 3. Concretely, the surface attached to
Z is C^2 / (Z^2 + Z Z^2), with the standard symplectic form.
A_2 is not compact. It has a boundary (the Baily-Borel, or toroidal,
boundary) reached by degenerating the lattice; Section 8 is about that.
1.4 SPECIAL SUBVARIETIES AND ZILBER-PINK
The special subvarieties of A_2 are the Shimura subvarieties together
with all their Hecke translates (a Hecke translate means: replace the
surface by an isogenous one with the transported polarization). The
ones we need:
- SPECIAL POINTS. The moduli points of surfaces isogenous to
E1 x E2 with BOTH E1 and E2 having complex multiplication. These are
zero-dimensional, hence codimension 3.
- E x CM CURVES. Fix a CM elliptic curve F. The surfaces isogenous to
E x F, with E an arbitrary elliptic curve, form a curve in A_2,
because E varies in a one-dimensional family. Every Hecke translate
of that curve is another such curve. These are the codimension-2
special subvarieties relevant here.
- E x E CURVES. Similarly, isogenous to E x E with E varying, plus
Hecke translates. Also codimension 2.
- HILBERT MODULAR SURFACES. These are codimension 1 (dimension 2).
ZILBER-PINK, in this setting, predicts: for a curve C contained in NO
proper special subvariety of A_2, the intersection of C with the union
of all special subvarieties of codimension at least 2 is finite.
Why "codimension at least 2": a curve and a codimension-1 subvariety in
a threefold meet, typically, in finitely many points already; if C is
Hodge generic it is not contained in that surface, so the intersection
is finite for trivial reasons. The interesting case is codimension 2: a
curve and a curve in a threefold have NEGATIVE expected intersection
number, so infinitely many accidental intersections would be a genuine
anomaly. Zilber-Pink says the anomaly cannot happen.
We therefore call C HODGE GENERIC when it lies in no proper special
subvariety, and we never need to discuss codimension-1 special
subvarieties after noting the preceding paragraph.
1.5 COMPLEX MULTIPLICATION
An elliptic curve F has complex multiplication (CM) when the algebra
End(F) (x) Q is an imaginary quadratic field K. Equivalently, F is
isomorphic to C / a for an ideal a of an order O in K, and then
End(F) = O up to the usual identification. Equivalently again, j_F is an
algebraic integer and [Q(j_F):Q] equals the class number h(O).
1.6 MIXED POINTS
Call a point s of A_2 MIXED when A_s is isogenous to a product of one CM
elliptic curve and one non-CM elliptic curve. Theorem 2.1 says that a
Hodge generic curve has only finitely many mixed points. The both-CM
points (the special points) are handled separately, and much more
easily, by Andre-Oort.
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2. THE TWO THEOREMS
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THEOREM 2.1 (main).
Let C be a reduced, irreducible, closed algebraic curve in A_2 defined
over Qbar, and suppose C is Hodge generic. Then the set
{ s in C(Qbar) : A_s is isogenous to E1 x E2, with at least one of
E1, E2 having CM }
is finite.
Remarks on the statement. The isogenies are isogenies of abelian
varieties; they are NOT required to respect the given polarizations.
Every isogeny degree, every imaginary quadratic field, and every
endomorphism order is allowed, including the case of two CM factors
(which is the trivial-to-handle part, see Section 12). There is no
assumption on the boundary of C, no integral base point, no condition on
reduction at any finite prime.
THEOREM 2.2 (Galois orbits).
Let C be as in Theorem 2.1 and let K be a number field over which C is
defined. There is a constant c > 0, depending only on C and K, such
that every mixed point s in C(Qbar) satisfies
#( Gal(Qbar/K) . s ) >= c * cplx(s)^(1/42).
Here the left side is the size of the Galois orbit of s, which equals
[K(s):K]. At a mixed point, let N(s) be the smallest degree of an
isogeny from an elliptic product to A_s; write N(s) = n^2 with n as in
Section 6. The CM factor of a product attaining the minimum is unique up
to isomorphism (Lemma 6.1). Then
cplx(s) = max{ N(s), |disc End(F)| }.
Remarks.
- The exponent 1/42 = 1/(2*21) is deliberately awful. What matters
for Section 12 is only that it is positive and explicit as a power.
Nothing here is optimised.
- The proof uses Siegel's class-number bound, which is ineffective.
Remark 5.2 describes an effective replacement for that one input
only. We make no effectivity claim for Theorem 2.1 or 2.2.
- Why Theorem 2.2 proves Theorem 2.1 is Section 12; it is the Pila-
Zannier strategy, with the counting theorem of Daw and Orr.
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3. THE PLAN OF THE PROOF
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STEP 1 (Section 6). At a mixed point, the product structure is canonical.
There are unique elliptic curves E (non-CM) and F (CM) inside A_s, and
the addition map E x F -> A_s is an isogeny of degree n^2 whose kernel
is the graph of an isomorphism theta : F[n] -> E[n]. The integer n is
intrinsic, and N(s) = n^2.
STEP 2 (Sections 7, 8). At a finite prime where E degenerates to a Tate
curve, the one-dimensional torus that appears in the degeneration of A_s
has a length, and that length is EXACTLY log|j_E|_v / n. The factor 1/n
is the surprise, and it is what makes the argument work at the boundary.
A fixed integral compactification of the moduli space measures that
length, up to two fixed constants, at every finite prime at once.
STEP 3 (Section 9). On a FIXED curve C, the height of j_E is controlled
by the height of the point on C. Step 2 converts that into
h(j_E) << n * (1 + sqrt D),
and the companion reverse inequality allows the coefficient to be
absorbed for n large. This is Proposition 9.1.
STEP 4 (Section 10). Repeat the non-CM factor. Build an abelian
threefold B isogenous to E x E x F, with a polarization balancing the
two very different period lengths of E. One period of B is then
essential, and the period theorem (Tool 2) gives
n^(4/3) << (b^2 + 1) * d * max{1, hF(B), log deg L}.
STEP 5 (Section 11). Feed the CM class-number bound, Step 3 and Step 4
into one another. The height and degree terms of B are bounded by a
power of d, and d is bounded by the size of the Galois orbit. The
result is
orbit size >> cplx(s)^(1/42).
STEP 6 (Section 12). Daw-Orr's counting theorem turns a positive power
orbit bound into finiteness; Andre-Oort handles the both-CM points.
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4. CONVENTIONS: HEIGHTS, PERIODS, ESSENTIAL PERIODS
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4.1 ENDOMORPHISMS. Endomorphism rings and Hom groups are geometric
(computed over Qbar) unless a ground field is explicitly mentioned.
4.2 HEIGHTS. All heights are absolute logarithmic heights. For x in a
number field k,
h(x) = sum over places v of k of [k_v : Q_v] / [k : Q] * log^+|x|_v.
Absolute values at finite places extend the standard p-adic ones on Q and
are NOT renormalized after a field extension; the weights
[k_v:Q_v]/[k:Q] already do the normalizing. Equivalently, the
archimedean contribution is the average over the complex embeddings,
with conjugate embeddings counted separately. With this normalization
h(x) does not change when the field of definition is enlarged.
4.3 FALTINGS HEIGHT. Write hF(A) for the stable Faltings height of an
abelian variety A, in the normalization of Gaudron-Remond. Put
hF^+(A) = max{1, hF(A)}. For a line bundle L on a g-dimensional abelian
variety B set
deg_L B = c_1(L)^g [B],
so in this convention a principal polarization has degree g! (for g = 2
that is 2, for g = 3 that is 6).
4.4 ELLIPTIC PERIODS. For an elliptic curve with its principal
polarization, choose a period basis e, e*tau with tau in the standard
closed fundamental region, and write y = Im(tau). Then
y >= sqrt(3)/2, ||e||^2 = 1/y, ||e*tau||^2 = |tau|^2/y << y.
The Fourier expansion of j, together with compactness away from the
cusp, gives
| log^+|j(tau)| - 2*pi*y | << 1.
The constants in these two displays are absolute. All other implicit
constants in this paper may depend on the fixed curve, the fixed family,
and the fixed ground field, but never on the varying point or on the
field of definition of that point.
4.5 ESSENTIAL PERIODS. Let B be a complex abelian variety and let
omega be a nonzero period of B, i.e. a nonzero element of Lie(B) which
lies in the period lattice. Call omega ESSENTIAL if omega lies in the
Lie algebra of no proper abelian subvariety of B. Equivalently: no
nonzero complex subtorus of B has tangent space containing omega.
For a polarized abelian variety (B,L) of dimension g over a number field
k, and an essential period omega at one chosen complex embedding,
Tool 2 below bounds (deg_L B)^(1/g) / ||omega||_L^2.
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5. THE TOOLBOX
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TOOL 1 (Faltings height: additivity and isogeny invariance). For abelian
varieties B1, B2 and an isogeny phi : B1 -> B2,
hF(B1 x B2) = hF(B1) + hF(B2),
| hF(B1) - hF(B2) | <= (1/2) * log(deg phi).
See Gaudron-Remond, Section 2.3. These are statements about the stable
height, so passing to a semistable extension changes nothing.
For an elliptic curve P one has
1 + h(j_P) ~ 1 + hF^+(P)
with absolute implied constants; see Silverman, Proposition 2.1, and the
normalization recorded in Loebrich, Proposition 3.1.
TOOL 2 (period theorem of Gaudron-Remond). Let (B,L) be a polarized
abelian variety of dimension g over a number field k, and let omega be
an essential period of B at one complex embedding. Then
(deg_L B)^(1/g) / ||omega||_L^2
<<_g [k:Q] * max{1, hF(B), log deg_L B}. (5.1)
The estimate allows an arbitrary polarization degree. Here is why the
field degree appears with exponent one, which is what we use. Let
delta_0(B_sigma, L_sigma) be the least norm of an essential period of B
at the embedding sigma, with value +infinity when there is none. The
essential-minimum form of their theorem says
(1/[k:Q]) * sum over sigma of (deg_L B)^(1/g) / delta_0(sigma)^2
<<_g max{1, hF(B), log deg_L B}.
At our chosen embedding sigma we have delta_0(sigma) <= ||omega||_L, so
that one summand, divided by [k:Q], is at most the left side of (5.1)
divided by [k:Q]. Keeping only that nonnegative summand gives (5.1).
TOOL 3 (family height bound). Let X -> S be a fixed principally
polarized abelian scheme over a smooth curve S defined over a number
field, and let Sbar be a smooth projective completion. Let H be an ample
divisor on Sbar and let h_H be shifted so that h_H >= 0. Then
hF^+(X_t) << 1 + h_H(t) for all t in S(Qbar). (5.2)
Proof. After a fixed finite cover and after restricting to a dense open
subset, choose a symmetric ample line bundle representing the
polarization together with theta data, as in Pazuki, Section 2.3, at one
fixed even theta level r >= 2. This is possible by choosing the data
over a finite extension of the function field and spreading out over the
open subset. The theta-null map is an algebraic map to a projective
space, and Pazuki's comparison of the theta height with the Faltings
height gives hF^+ << 1 + h_theta, with constants depending only on the
dimension and on the fixed theta level.
The theta-null map extends to the smooth projective covering curve. Let
p be the covering map and let T be the pullback of the hyperplane
divisor. Choose an integer a with a*p^*H - T ample; its height is
bounded below, so the height machine gives
h_theta <= a * h_H(p(t)) + O(1).
That proves (5.2) after lifting t, and the stable Faltings height is
unchanged by the lift. The finitely many points omitted while choosing
the theta data are absorbed by enlarging the constant. []
TOOL 4 (CM class numbers: Siegel). For the maximal order O_{D_0} in an
imaginary quadratic field of absolute discriminant D_0,
h(O_{D_0}) >> D_0^(1/4).
The implied constant is absolute but ineffective. This is the only
ineffective input in the paper.
Now the one computation we need about CM elliptic curves. It combines
Tools 4 with the conductor formula and the geometry of the period
lattice.
LEMMA 5.1 (CM estimates). Let k be a number field and let F/k be a CM
elliptic curve. Put O = End(F), D = |disc O|, d = [k:Q]. Then
D << d^4, h(j_F) << 1 + sqrt(D). (5.3)
At any complex embedding choose a reduced period basis f, f*tau_F and
put y_F = Im(tau_F). Then
b := [ Lambda_F : O*f ] = sqrt(D) / (2*y_F) is a positive integer,
y_F <= sqrt(D)/2, b << sqrt(D). (5.4)
Proof. The degree [Q(j_F):Q] is the class number h(O), and j_F is an
algebraic integer (Cox, Theorem 11.1 and Proposition 13.2). Write
D = f_0^2 * D_0 with D_0 the absolute discriminant of the maximal order.
By Tool 4, h(O_{D_0}) >> D_0^(1/4). The conductor formula (Cox,
Theorem 7.24) says h(O) is h(O_{D_0}) multiplied by
f_0 / [ O_{D_0}^x : O^x ] * prod over p | f_0 of ( 1 - chi(p)/p ),
where chi is the quadratic character of the field of discriminant -D_0,
given by the Kronecker symbol, and where the product is over rational
primes. The factor f_0 / [units] is at least f_0/3, and each Euler
factor is at least 1 - 1/p, so the whole product is
>= (f_0/3) * prod_{p | f_0} (1 - 1/p) = phi(f_0)/3 >> sqrt(f_0),
using the elementary bound phi(f_0) >= sqrt(f_0/2). Therefore, for every
conductor,
d >= h(O) >> D^(1/4),
which is the first half of (5.3). The implied constant may be
ineffective.
For the period statement: O is a lattice in C of covolume sqrt(D)/2, the
period lattice Lambda_F = Z + Z*tau_F has covolume y_F, and O*f is
contained in Lambda_F, so the index is the quotient of the covolumes,
which is the first formula in (5.4). The lower bound y_F >= sqrt(3)/2
gives y_F <= sqrt(D)/2 and b << sqrt(D).
For the height: at every archimedean embedding, the Fourier estimate of
4.4 bounds log^+|j_F| by 2*pi*y_F + O(1) = O(1 + sqrt(D)). Since j_F is
an algebraic integer, there is no finite-place contribution at all, so
h(j_F) is just the average of these archimedean terms. []
REMARK 5.2 (an effective replacement for Tool 4 alone). The Dirichlet
assertion of the quasi-Riemann-hypothesis preprint supplies a zero-free
half plane Re(s) > 7/8, which in turn supplies the short real zero-free
interval used in Chen's proof of the Siegel-Tatuzawa-Hoffstein theorem.
The outcome is L(1, chi) >> 1/log(2D_0) with an EFFECTIVE absolute
constant, after treating finitely many small conductors by hand. The
Dirichlet class-number formula then gives
h(O_{D_0}) >> sqrt(D_0)/log(2 D_0) >> D_0^(1/4)
effectively, and the conductor estimate above then gives d >> D^(1/4)
with an effective constant for every order. This remark makes only the
CM discriminant estimate effective. It says nothing about effectivity of
the constants in Theorems 2.1 and 2.2.
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6. CANONICAL GLUING: THE INTEGERS n AND theta
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At a mixed point the product decomposition of A_s is not a choice. It is
forced by the surface itself. This section makes that precise, and it
also makes precise how a Galois group can act on it.
LEMMA 6.1 (canonical product isogeny). Let (A,lambda) be a principally
polarized abelian surface over a field of characteristic zero, and
suppose A is isogenous (geometrically) to a product E x F of a non-CM
elliptic curve E and a CM elliptic curve F. Then:
(a) There are unique elliptic subvarieties E, F of A, with E non-CM
and F CM. They are defined over the field of definition of A.
(b) There is a positive integer n, and addition of the two subvarieties
is an isogeny
phi : E x F -> A,
of degree n^2, whose pullback of lambda is n times the product
polarization, and whose kernel is the graph of an isomorphism
theta : F[n] -> E[n].
This theta is defined over the field of definition of A.
(c) The least degree of an isogeny from any elliptic product P x Q to A
is n^2, and for any such minimal isogeny the factors (P, Q) are
isomorphic to (E, F) in some order.
Proof.
STEP A: uniqueness. The two simple isogeny factors E and F are not
isogenous: a nonzero homomorphism E -> F would have finite kernel (both
curves being simple), hence would be an isogeny, and then E would have
CM too, since End(F) contains an imaginary quadratic field. So
Hom(E, F) = 0 and Hom(F, E) = 0. Consequently every abelian subvariety
of A is isogenous to one of 0, E, F, A.
Now let E1, E2 be two abelian subvarieties of A both isogenous to E.
Their sum E1 + E2 is an abelian subvariety of A whose isogeny class
contains E twice. The only subproduct of E x F containing E twice is E
itself, so E1 + E2 is isogenous to E. Since E1 and E2 have the same
dimension as E, this forces E1 = E2. The subvariety E is therefore
unique, and similarly F. A geometrically unique subvariety is stable
under the absolute Galois group, hence is defined over the field of
definition of A. That is (a).
STEP B: the multipliers. Let phi : E x F -> A be the addition map. The
pullback phi^* lambda is a product polarization; write its two elliptic
multipliers as n_E and n_F, both positive integers. In the degree
convention of 4.3, the product of the degrees is
deg( phi^* lambda ) = 2 * n_E * n_F, deg( lambda ) = 2,
so the kernel has order n_E * n_F.
STEP C: the kernel is a graph. Let phi^ : A -> E x F be the dual
isogeny. By construction of the dual, phi^ composed with phi is the
multiplication by the diagonal matrix diag(n_E, n_F) on E x F. Hence, for
(x, y) in ker phi,
n_E * x = 0 and n_F * y = 0,
so ker phi is contained in E[n_E] x F[n_F].
The two projections of ker phi to its factors are injective. Indeed, if
phi(x, 0) = 0 then x maps to zero in A, so x = 0 because the inclusion
E -> A is injective; and likewise for F. So
n_E * n_F = #ker phi <= #E[n_E] = n_E^2,
which gives n_F <= n_E, and symmetrically n_E <= n_F. Put n_E = n_F = n.
Then #ker phi = n^2 = #E[n] = #F[n], and each projection is an injective
map between groups of the same size, so each is an isomorphism. A
subgroup of F[n] x E[n] projecting isomorphically onto both factors is
the graph of an isomorphism theta : F[n] -> E[n]. This gives (b), apart
from the field of definition of theta, which holds because ker phi is
defined over the field of A.
STEP D: minimality. Let psi : P x Q -> A be any isogeny from an elliptic
product. The image of P is an abelian subvariety of A isogenous to the
simple curve P, hence by Step A it is E or F; likewise the image of Q is
the other one. So psi factors as
P x Q -> E x F -> A,
and deg psi = deg(P x Q -> E x F) * n^2 >= n^2. Equality forces the
first map to be an isomorphism of polarized products. That is (c). []
REMARK 6.2 (how much Galois does the CM factor cost). The absolute
Galois group acts on the endomorphism ring of a CM elliptic curve F
through its action on the imaginary quadratic field End(F) (x) Q, whose
automorphism group has order two. Consequently, if A is defined over a
number field k1, then ALL endomorphisms of F are defined over a field
extension of k1 of degree at most two. Defining the graph theta itself
costs nothing: theta comes from the kernel of phi, which is already
defined over k1.
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7. DEGENERATION AT A FINITE PRIME: THE LENGTH IS log|j_E|/n
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The whole boundary argument rests on one local computation, which is
surprising the first time: the index n, which makes the product isogeny
fatter, makes the degeneration SHORTER by exactly the same factor.
THE LOCAL PICTURE. Let a principally polarized abelian variety over a
complete discretely valued field have split semiabelian reduction. Its
uniformization is
(an extension of a good-reduction abelian variety by a split torus T)
modulo a period lattice Y.
Let X be the character lattice of T. A polarization gives a map Y -> X,
and that map is an isomorphism in the principally polarized case. The
degeneration data carry a functorial valuation pairing
Y x X -> R,
computed with the extended absolute value. In toric rank one, the two
generators matched by the principal polarization pair to a single
positive number, and that number is what we call ell(A).
LEMMA 7.1 (reciprocal length). In the situation of Lemma 6.1 over a
number field, let v be a finite place. If E has potential good reduction
then so does A. Otherwise, after passing to an extension over which the
reduction is split semiabelian,
ell(A) = log|j_E|_v / n. (7.1)
Here |j_E|_v is the absolute value extended from the original field, and
the formula is unchanged by the extension.
Proof.
STEP A: F has potential good reduction. A CM elliptic curve has
potential good reduction (Serre-Tate, Theorem 6(a)); for an elliptic
curve this also follows directly, because j_F is an algebraic integer,
so its reduction is an element of the valuation ring, so the curve does
not degenerate to a nodal cubic with |j| < 1. Hence, after a finite local
extension, F has genuine good reduction.
STEP B: E has good or split Tate reduction. After that extension, E has
either good reduction or split multiplicative (Tate) reduction (Tate,
Theorem 1 and applications). Potential good reduction and rational toric
rank are invariants of the isogeny class, so we may compute in whichever
model of E we like. It remains to find the length in the Tate case.
STEP C: the torus map is an isomorphism. Semiabelian uniformization is
functorial (Lan, Corollary 4.5.5.5), so phi : E x F -> A gives a map of
one-dimensional tori. This map has no nontrivial kernel: a nontrivial
kernel would contain a nontrivial root of unity, and torsion of the torus
injects into the Tate elliptic curve (whose period lattice is torsion
free). Such a root of unity would give a nonzero point of ker phi lying
in E x {0}, contradicting the graph description in Lemma 6.1. So the
torus map is an isomorphism of one-dimensional tori. All of this happens
on characteristic-zero generic fibres; in particular it does not matter
whether v(n) = 0 or not.
STEP D: the multiplier n. Let Y', X' be the lattices attached to the
product E x F, polarized by the product principal polarization. Pulling
back lambda means that the composite
Y' -> Y_A -> X_A -> X'
is multiplication by n times the product principal polarization. The
outer two arrows are the principal-polarization isomorphisms, and the
middle arrow phi_X^* is an isomorphism by Step C. Therefore the lattice
map Y' -> Y_A has absolute multiplier n.
STEP E: functoriality of the pairing. The valuation pairing is functorial,
so for y in Y' and x in X',
< phi_Y y , x > = < y , phi_X^* x > .
Taking the generators matched by the principal polarizations, the map
phi_Y has multiplier n, so
n * ell(A) = ell(E x F).
For the Tate parameter q_E of E, the right side equals -log|q_E|_v, and
in the Tate model the j-invariant satisfies j = q_E^(-1) times a unit
(Tate, Equation 18), so -log|q_E|_v = log|j_E|_v. This is (7.1). []
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8. THE BOUNDARY OF THE MODULI SPACE
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We have a length. We need the moduli space to measure it for us, and we
need the measurement to behave uniformly when the ground field grows.
This section supplies both, plus the two archimedean facts used later.
THE INTEGRAL MODEL. Fix a level m >= 3. There exists a smooth
projective toroidal compactification of the Siegel moduli scheme over
Z[zeta_m, 1/m], with reduced boundary divisor D_m, and that boundary is
an effective relative Cartier divisor. We will only ever use m = 3 and
m = 4, and those two levels together cover every finite prime, because
a prime not dividing m is allowed in Lemma 8.1 below and 3*4 = 12 has no
prime factor missed.
LEMMA 8.1 (integral boundary comparison). Fix the level m and the
compactification above. There are constants c_m, C_m > 0 depending only
on those choices, with the following property. Let k_v / Q_p be a finite
extension with p not dividing m, and let A / k_v be a principally
polarized abelian surface with full level m. Write lambda_{D_m,v}(A) for
the model local height of the boundary divisor at the point.
(i) If A has potential good reduction, then lambda_{D_m,v}(A) = 0.
(ii) If A has potential split semiabelian reduction of toric rank
one, then
c_m * ell(A) <= lambda_{D_m,v}(A) <= C_m * ell(A).
The quantity ell(A) is computed after an extension giving split
semiabelian reduction, and both sides are unchanged by any further local
extension.
Proof. This is a chart computation in the integral toroidal
compactification. The construction is due to Faltings-Chai; we use Lan,
Construction 6.3.1.1 and Theorem 6.4.1.1, with projectivity supplied by
Lan, Theorem 7.3.3.4.
STEP A: normal coordinates. By Lan, Theorem 6.4.1.1 parts (2) and (5),
the completed neighbourhood of a stratum is the prescribed torus
embedding over an abelian-scheme torsor, which itself sits over a finite
etale cover of a lower-dimensional moduli space. By part (3), the
boundary is the reduced relative Cartier normal-crossings divisor of that
torus embedding. Its strata pull back scheme-theoretically to the
toroidal strata, and their normal coordinates are the toric monomials
(Lan, Proposition 6.3.1.6(1)).
STEP B: the boundary ideal. Trivialize the torus torsor on a smooth cone
chart, and extend the primitive ray generators to a lattice basis. In the
resulting coordinates q_1, ..., q_r the reduced boundary ideal is
( q_1 * q_2 * ... * q_r ),
which is the relative toric divisor of Lan, Theorem 6.1.2.8(5).
STEP C: there is no extra coordinate from the base. In a rank-one
degeneration the abelian quotient has good reduction, so its moduli point
lies in the INTERIOR of that lower-dimensional base. The extension
coordinates live on the abelian-scheme torsor and, after a local
extension, extend over the valuation ring by properness. This holds at
ordinary and at supersingular points of the base alike. So nothing extra
from the base or from the extension contributes to the boundary
valuation.
STEP D: exactly one monomial counts. The valuation form lies on a
rank-one ray of the cone. If a rank-one positive semidefinite form is a
positive sum of such forms, then each summand vanishes on its own kernel
and lies on the same ray. Hence the ray is a face of every cone that
contains the form. In a smooth cone chart, only the primitive normal
monomial q_c has positive valuation; the other normal monomials are
units. Locally the reduced boundary ideal is therefore (q_c) up to a
unit. There is no vertical factor, because we are working with the
scheme-theoretic toric boundary ideal in a relative normal-crossings
chart over Z[zeta_m, 1/m], rather than with a comparison that only holds
on the generic fibre.
STEP E: matching with ell(A). Lan, Propositions 6.2.5.8(3), 6.2.5.11 and
6.3.1.6(4), identify the tautological ideal-valued toric pairing with the
pairing of the semiabelian degeneration. So -log|q_c| equals ell(A) times
a fixed rational factor which compares the primitive normal character with
the rank-one polarized cusp lattice. Those lattices, and their indices,
are part of the fixed level datum; they do not depend on the isogeny used
to present A, nor on the ramification of the field. There are finitely
many cusp types at a fixed level, so there are just two comparison
constants, c_m and C_m. Computing with the extended absolute value keeps
the values unchanged under any further local field extension.
STEP F: good reduction, and small primes. If A has good reduction its
point is in the interior and every boundary monomial is a unit, so the
boundary local height is zero. Finally, the integral Siegel model used
here only needs m to be invertible in the base, including at residue
characteristic two. This is the good-prime condition of Lan, Definition
1.4.1.1: for the self-dual symplectic datum over Z the algebra is Q, the
order is Z, and there is no type-D or discriminant exclusion. Levels 3
and 4 handle every prime. []
UNIFORMITY IN THE HEIGHT MACHINE. We need one bookkeeping statement.
An M_{K_0}-BOUND is a family of constants c_v, zero at all but finitely
many finite places v of K_0, inherited at places above v by extending its
absolute value, with the weighted sum bounded independently of any finite
extension of K_0. That is precisely what survives normalization in 4.2.
Now fix a smooth curve S over K_0 with a principally polarized abelian
scheme of relative dimension two and level divisible by 12, and a smooth
projective completion Sbar. The level-forgetting maps extend by
properness to morphisms
f_m : Sbar -> A_2bar_m (m = 3, 4),
and we let Delta be the reduced inverse image of the boundary. Its
support does not depend on m, because it is characterized by: the
potential toric rank is nonzero.
If Delta is nonempty, write f_m^* D_m = Delta + R_m with R_m effective.
Local heights of effective divisors on a proper variety are bounded
below, so functoriality gives
lambda_{Delta,v}(t) <= lambda_{D_m,v}(f_m(t)) + O_v(1). (8.1)
If Delta is empty, the generic pullback divisor is zero, so instead
lambda_{D_m,v}(f_m(t)) = O_v(1). (8.2)
Both errors are M_{K_0}-bounds, so their weighted sums are bounded once
and for all.
Here is the model justification that also handles the finitely many bad
primes of the curve and of its maps. Over O_{K_0}[1/m], take a proper
model of Sbar, then the closure of the graph of f_m inside its product
with the fixed integral toroidal compactification. If needed, pass to a
fixed further model on which the divisors involved are Cartier. Models of
the same generic divisor differ by vertical divisors at finitely many
primes, and an effective vertical Cartier divisor V is bounded above by a
fixed multiple of the full special fibre: locally a power of the
uniformizer lies in the ideal of V, and quasi-compactness supplies a
common power. Taking positive and negative bounds on the vertical
discrepancy shows that changing the model perturbs the local height by a
bounded amount in the extended absolute value. That is the asserted
uniformity, even while the point and the ramification of its field vary.
THE ARCHIMEDEAN BOUNDARY ESTIMATE. At a complex point, let rho(A) be
the length of the shortest nonzero period of A for its principal
polarization.
LEMMA 8.2. If Delta is nonempty, then at each archimedean place of K_0,
lambda_Delta(t) << 1 + rho(A_t)^(-2).
Proof. Work near one point of Delta, with disk parameter z. After a
fixed ramified base change, monodromy on H_1 is unipotent, with logarithm
N satisfying N^2 = 0 (Grothendieck, SGA7, the monodromy description of a
semistable degeneration of abelian varieties).
STEP A: a convenient basis. The image of N is isotropic. Choose a
rational Lagrangian subspace containing it and contained in ker N, and
adapt an integral symplectic basis to the primitive lattice of that
Lagrangian. On the universal cover of the punctured disk, the period
matrix Z then has monodromy
Z |-> Z + M, M in Sym_2(Z).
STEP B: M is positive semidefinite. For u in Z^2 the function
q_u = exp( 2 pi i * u^t Z u ) is single-valued and bounded by 1 in
absolute value, because Im Z is positive definite. It therefore extends
across z = 0, and its vanishing order there is u^t M u. Since the order is
nonnegative for every u, M is positive semidefinite. Writing
q_u = z^(u^t M u) times a holomorphic unit, and applying this to the
finitely many vectors e_i and e_i + e_j, shows
Y := Im Z = ( -log|z| / (2 pi) ) * M + R(z), (8.3)
where R extends harmonically across zero.
STEP C: the kernel block stays positive definite. On ker M, the form
R(0) is positive definite. For any nonzero real v in that kernel, the
function z |-> v^t R(z) v equals v^t Y v, which is positive on the
punctured disk. If its value at zero were zero, the minimum principle
would force this harmonic function to vanish identically. If M were
zero, that same argument plus the holomorphic extension constructed in
Step B would extend Z to the interior of Siegel space, and uniqueness of
extension to the separated toroidal compactification would contradict
the choice of a boundary point. Hence M is not zero.
STEP D: a short period. Choose a nonzero integral w in the image of M,
which is possible because M is integral. With respect to the orthogonal
decomposition im M plus ker M, the im M block of Y grows like a positive
definite matrix times -log|z|, the off-diagonal blocks are bounded, and
the kernel block stays positive definite. The Schur complement formula
gives
w^t Y^(-1) w = O( 1 / (-log|z|) ).
That quantity is exactly the squared norm of the period encoded by w in
the lattice Z^2 + Z Z^2, so
rho(A_t)^2 << 1 / (-log|z|), equivalently
-log|z| << rho(A_t)^(-2).
The boundary local height equals a fixed multiple of -log|z| up to a
bounded term, also after the fixed ramification. Finitely many such
disks, plus compactness on their complement, finish the proof. Both ranks
one and two for M are included. []
THE SHORTEST PERIOD AT A MIXED POINT.
LEMMA 8.3. At a mixed point, choose reduced elliptic periods at a complex
embedding, and write y_E = Im(tau_E) and y_F = Im(tau_F). Then
rho(A)^(-2) <= n * y_F + y_E / n, rho(A)^2 <= n / y_E.
Proof. Use phi : E x F -> A to identify the Lie algebras. The period
lattice of A is an overlattice of Lambda_E plus Lambda_F, contained in
their 1/n multiple, and its Hermitian form is n times the product form.
Its intersection with Lie(E) is exactly Lambda_E: a lattice vector with
zero F-coordinate has its F-part in Lambda_F intersected with
(1/n)Lambda_F, which is zero torsion, so the corresponding element of
ker phi has trivial F-component, hence is zero.
A lattice vector with nonzero F-projection has F-part at least the first
reduced period f of F in size, so its squared norm is at least
n * (1/n^2) / y_F = 1/(n y_F). A nonzero lattice vector with zero
F-projection has E-part a nonzero element of Lambda_E, of squared norm at
least 1/y_E, giving squared norm at least n/y_E, and the first reduced
period of E attains that. The two assertions follow. []
---------------------------------------------------------------------------
9. THE HEIGHT BOUND ON A CURVE
---------------------------------------------------------------------------
Now fix ONE curve, once and for all. The rest of the paper is estimates
on that curve with constants allowed to depend on it.
Let S be a smooth curve over a number field K_0 with a principally
polarized abelian scheme of relative dimension two and level divisible by
12, and let Sbar be its smooth projective completion, as in Section 8.
PROPOSITION 9.1 (uniform height bound on a fixed curve). There are
constants c > 0 and n_0 > 0, depending only on the family, such that
every mixed point t in S(Qbar), with the notation of Lemma 6.1 and with
D = |disc End(F)|, satisfies
n >= n_0 implies h(j_E) <= c * n * ( 1 + sqrt(D) ).
No Hodge genericity is needed here. The proof has two cases, and it uses
everything in Sections 7 and 8.
Proof.
CASE 1: the boundary is empty. At every complex embedding of K_0, the
image of Sbar is compact inside the interior of the moduli space. The
shortest period length therefore has a positive lower bound on that
image, say rho(A_t) >= r for a fixed r > 0. By Lemma 8.3,
rho(A)^2 <= n/y_E, so
y_E <= n / r^2
at every embedding of a field of definition of t. Combined with the
Fourier estimate |log^+|j| - 2 pi y| << 1 of 4.4, and with (8.2) summed
with the normalized weights, this gives h(j_E) << n. Done.
CASE 2: the boundary is nonempty. Choose h_Delta shifted so that
h_Delta >= 0; it is an ample height, because Delta is a nonzero
effective divisor on a projective curve.
STEP A: the archimedean contribution. Lemma 8.2 and Lemma 8.3, together
with the bound y_F <= sqrt(D)/2 from (5.4), give at the archimedean places
lambda_Delta(t) << 1 + n * sqrt(D) + y_E / n.
Replace y_E using 4.4, add (8.1) at the finite places, and sum with the
normalized weights. The M_{K_0}-bounds of Section 8 give a constant
independent of the field. The outcome is
h_Delta(t) <= B * ( 1 + n*sqrt(D) + h(j_E)/n ) (9.1)
for a fixed B > 0. The key point is the coefficient 1/n in front of
h(j_E): it is exactly the reciprocal length of Lemma 7.1 that produced it.
STEP B: the reverse inequality. By Tool 1,
hF(E) + hF(F) <= hF(A) + log n.
The comparison 1 + h(j_E) ~ 1 + hF^+(E) of Tool 1 and the family height
bound (5.2) then imply
h(j_E) <= A_0 * ( 1 + h_Delta(t) + log n ) (9.2)
for a fixed A_0 > 0.
STEP C: absorption. Substitute (9.1) into (9.2). The coefficient of
h(j_E) on the right side becomes A_0 B / n. For n >= 2 A_0 B it can be
moved to the left side. Since log n << n, the result is
h(j_E) << n * ( 1 + sqrt(D) ),
as claimed. []
---------------------------------------------------------------------------
10. AN ESSENTIAL PERIOD ON A THREEFOLD: THE EXTRA POWER OF n
---------------------------------------------------------------------------
Section 9 says h(j_E) grows at most like n. Section 5 says D grows at
most like d^4. If we could only convert those two facts into a bound on
n, we would be done, and we cannot: something must force n itself to be
small. That something is the period theorem, applied to a purpose-built
abelian threefold.
The construction repeats the non-CM factor. A period vector on E x E x F
can then see both independent periods of E, so no integer linear
combination can make it vanish, and the polarization can be weighted to
balance the two very different lengths 1/y and y of E.
SETUP. Let k be a number field containing the definitions of E, F, theta
and ALL endomorphisms of F. Put d = [k:Q] and fix one complex embedding.
Take a reduced basis e_1, e_2 of the period lattice Lambda_E, and a first
reduced period f of the CM curve F. Use b and y_F from Lemma 5.1, and put
y = y_E.
The two squared period lengths of E have sizes 1/y and y. Weighting the
first copy of E by about y^2 makes their contributions comparable.
Weighting F by about y then makes the cube root of the polarization
degree grow at the same rate as the squared norm. What remains is a net
gain of n^(4/3), supplied by the quotient construction.
PROPOSITION 10.1 (the threefold). There is a polarized abelian threefold
(B, L) over k, isogenous to E x E x F, such that
n^(4/3) << (b^2 + 1) * d * max{ 1, hF(B), log deg_L B }, (10.1)
and
hF(B) <= 2*hF(E) + hF(F) + log n,
deg_L B = 6 * n^4 * u * v, u = ceil(y^2), v = ceil(y). (10.2)
Proof.
STEP A: manufacturing the small endomorphisms. For i = 1, 2 choose
f_i in Lambda_F whose class modulo n*Lambda_F is theta^(-1)( e_i/n ),
that is, theta( f_i/n ) = e_i/n as points of E[n]. Since
b = [Lambda_F : O*f], we have b*Lambda_F contained in O*f = End(F)*f,
so there is beta_i in End(F) with
beta_i * f = b * f_i. (10.3)
Nothing requires beta_i to act invertibly on F[n].
STEP B: the quotient. Define
Gamma = { ( theta(beta_1(p)), theta(beta_2(p)), p ) : p in F[n] },
B = (E x E x F) / Gamma.
All maps in this display are defined over k. The endomorphisms beta_i are
elements of End(F), all of which lie in k by the choice of k, and theta
is defined over k by Lemma 6.1. The last-coordinate projection is inverse
to the displayed graph embedding, as a map of group schemes; so Gamma is
identified with F[n], and the quotient map
pi : E x E x F -> B
has degree n^2. There is no coprimality requirement on n, b, beta_1,
beta_2 at any point of this argument.
STEP C: the dual and the polarization. Since Gamma is killed by n, the
quotient property factors the multiplication by n map, over k, as
E x E x F --pi--> B --psi--> E x E x F, psi o pi = [n].
Since deg[n] = n^6 and deg(pi) = n^2, we get deg(psi) = n^4.
Give the product the line bundle
M = O_E(u[0]) box O_E([0]) box O_F(v[0]), L = psi^* M.
These are ample and defined over k. The integers u, v were chosen using
one embedding, but that choice needs no extension of the field. By 4.3,
deg_M(E x E x F) = 3! * u * 1 * v = 6uv, hence deg_L B = 6 n^4 u v. The
height bound is Tool 1 applied to the isogeny pi: deg pi = n^2 gives
hF(B) <= hF(E x E x F) + log n. That is (10.2).
STEP D: a period of B. In the product Lie space, the vector
omega = ( b*e_1 , b*e_2 , f ) / n
lies in the period lattice of B. Indeed, (10.3) gives
theta(beta_i(f/n)) = b*e_i/n, and that is precisely the membership
condition for being the lift of an element of Gamma.
STEP E: omega is essential. Suppose omega lay in the Lie algebra of a
proper abelian subvariety W of B. The identity component of pi^(-1)(W)
would then be a proper abelian subvariety of E x E x F whose Lie algebra
contains omega. By semisimplicity up to isogeny, such a subvariety is
annihilated by a nonzero homomorphism either to E or to F.
A homomorphism E x E x F -> E is an integer row (a_1, a_2, 0), because
End(E) = Z and Hom(F, E) = 0. Its value on omega would be
(a_1 b e_1 + a_2 b e_2)/n, and this vanishes only if a_1 = a_2 = 0,
because e_1 and e_2 are linearly independent over Z (indeed over R,
since tau_E is not real).
A homomorphism E x E x F -> F is (0, 0, beta) with beta in End(F), and
its value on omega is beta(f)/n, which is nonzero whenever beta is
nonzero, because f generates Lambda_F as an O-module.
A rational homomorphism can be cleared of denominators, which changes
nothing. Hence omega is essential. (Remark: this is where TWO copies of E
are needed. With one copy, the row (1, 0) would have killed the vector.)
STEP F: the norm. The identity pi^* L = [n]^* M gives
||omega||_L^2
= b^2 * u * ||e_1||^2 + b^2 * ||e_2||^2 + v * ||f||^2
<< (b^2 + 1) * y.
Here the bounds are: 1/y for ||e_1||^2, |tau_E|^2/y << y for ||e_2||^2,
u/y <= y + 1/y, v/y_F << y, all from 4.4. The factor n^2 in the
pulled-back metric has cancelled the denominator n^2 in the squared norm.
STEP G: the payout. We have (uv)^(1/3) >= y, so
(deg_L B)^(1/3) 6^(1/3) * n^(4/3) * (uv)^(1/3)
--------------- = ------------------------------- >> n^(4/3)/(b^2+1).
||omega||_L^2 ||omega||_L^2
Now apply Tool 2 to this essential period of a three-dimensional
polarized abelian variety. That is (10.1). []
REMARK 10.2 (why the beta_i do not spoil the bound). The large
endomorphisms beta_i never enter the estimate. Their job is to put a
specified vector into the quotient lattice, not to bound the degree of a
homomorphism. The projection of Gamma to F[n] fixes the order of Gamma
even if either beta_i has a large kernel on F[n].
---------------------------------------------------------------------------
11. GALOIS ORBITS: cplx(s)^(1/42)
---------------------------------------------------------------------------
We now prove Theorem 2.2. Fix C as in Theorem 2.1 and a number field K
over which C is defined.
STEP A: reducing to a smooth family. Pass to a component of a full-level
cover with level divisible by 12, and normalize that component. After
enlarging K once and for all to a number field K_0, this gives a smooth
curve S over K_0 with a family as in Section 9, and a map S -> C of fixed
bounded degree over a dense open subset of C. Further fixed covers or
restrictions used to describe the family have the same property. Only
finitely many points of C are omitted, and this is compatible with C
being singular.
STEP B: lifting a mixed point. Let s in C(Qbar) be a mixed point in the
dense open. Choose a lift t of s on S. Then
[K_0(t):Q] << [K(s):K],
because the finite fibre has bounded degree and K_0/K is fixed. By Lemma
6.1, E, F and theta are defined over the field of the fibre. By Remark
6.2, after an extension of degree at most two, all CM endomorphisms are
defined too. Let k be the resulting field and put d = [k:Q]. Then
d << [K(s):K]. (11.1)
Everything reduces to bounding max{n^2, D} by a fixed power of d.
STEP C: the estimates, for n large. Assume n >= n_0, the threshold of
Proposition 9.1. Lemma 5.1 gives D << d^4 and b^2 + 1 << d^4.
Proposition 9.1 gives h(j_E) << n(1 + sqrt D) << n d^2. Lemma 5.1 gives
h(j_F) << 1 + sqrt D << d^2. In summary,
D << d^4, b^2 + 1 << d^4, h(j_E) << n d^2, h(j_F) << d^2. (11.2)
STEP D: the threefold's size. Let (B, L) be the threefold of Proposition
10.1, with k now equal to the field of Step B. Using Tool 1 in the form
1 + h(j_E) ~ 1 + hF^+(E), and (11.2),
hF(B) <= 2*hF(E) + hF(F) + log n << n d^2.
For the polarization degree, use 4.4 at the chosen embedding. A single
archimedean contribution to an absolute height is at most d times the
total, so
y << 1 + d * h(j_E) << n d^3.
Then log(6 n^4 u v) << 1 + log n + log d << n d^2, because u << y^2 + 1
and v << y + 1. So the maximum in (10.1) is << n d^2.
STEP E: solving for n. Insert the last two paragraphs into (10.1):
n^(4/3) << d^4 * d * n d^2 = n d^7.
Dividing by n and cubing,
n << d^21, n^2 << d^42. (11.3)
STEP F: the small n. If n < n_0 then D << d^4 << d^42 and n^2 < n_0^2,
so after increasing the constant the same bound holds. In all cases,
cplx(s) = max{ n^2, D } << d^42 << [K(s):K]^42,
where the last step is (11.1) and Lemma 6.1 (the minimum degree equals
n^2, so cplx(s) is exactly max{n^2, D}). Taking 42nd roots,
#( Gal(Qbar/K) . s ) = [K(s):K] >> cplx(s)^(1/42).
The finitely many mixed points omitted in Step A are included by
reducing the positive constant. Theorem 2.2 is proved. []
---------------------------------------------------------------------------
12. FINITENESS ON THE ORIGINAL CURVE
---------------------------------------------------------------------------
We now go from a power orbit bound to finiteness. This step is due to
Daw and Orr, and it is the only place where the Zilber-Pink machine is
used.
12.1 THE COUNTING IMPLICATION. Daw and Orr prove (Theorem 1.2 of DO):
a Hodge generic curve in A_2 has finitely many mixed E x CM points,
provided their Galois orbits satisfy a power lower bound in the relevant
special curve complexity. Their complexity is the maximum of
(a) the minimal degree of an isogeny from an elliptic product to A_s,
(b) the discriminant of the endomorphism ring of the CM factor in a
product attaining that minimal degree.
By Lemma 6.1 this is exactly cplx(s), and by that same lemma the
minimal product is canonical. Their alternative complexity, used in the
proof, is polynomially comparable to this one (DO, Section 3, especially
Lemmas 3.1 and 3.6). So Theorem 2.2 supplies exactly the hypothesis.
There is no boundary hypothesis in this implication.
12.2 A POINT OF CARE. We use the CORRECTED version of DO. Its Lemma 2.3
verifies the Cartan-stability condition required by the corrected
reduction-theory input of Orr and Schnell. Without that verification the
counting theorem would not apply.
12.3 THE MIXED POINTS ARE ALGEBRAIC. In the Daw-Orr formulation the mixed
points are algebraic points. Here they are, by an argument independent of
the orbit bound. A mixed complex point s lies on an E x CM special curve
defined over Qbar: start with the diagonal product locus in which the CM
coordinate is fixed, and use the polarized isogeny of Lemma 6.1 to pass
through a Hecke correspondence, which is defined over Qbar. Such a
special curve meets C in a zero-dimensional set, because C is Hodge
generic. Those intersection points are therefore algebraic. The elliptic
subvarieties and their homomorphisms are then available over Qbar, by
invariance of homomorphisms of abelian varieties under algebraically
closed extension of the base field. So the complex and the algebraic
mixed loci coincide for our curve.
12.4 FINITELY GENERATED FIELDS. The Daw-Orr hypothesis is stated for a
curve over Qbar and also in a finitely generated field version. The
orbit bound supplies that version too. Given a finitely generated field
of definition L contained in C, let K be a number field of definition of
C, put M = L K, and put K' = M intersect Qbar. Then K' is a number field
containing K, and M/K' is regular. For an algebraic point s, linear
disjointness shows that its Aut(C/M)-orbit has cardinality [K'(s):K'].
Theorem 2.2 applies over K'. Since Aut(C/M) is a subgroup of Aut(C/L),
the L-orbit is at least as large, so the required lower bound holds for
it as well.
12.5 THE BOTH-CM POINTS. Suppose A_s is isogenous to E_1 x E_2 with BOTH
factors having CM. Then the product is of CM type, hence so is every
abelian surface isogenous to it, and so the moduli point s is a special
point of A_2. Andre-Oort for A_2, proved by Pila and Tsimerman (and
contained in Tsimerman's theorem for A_g), says that a Hodge generic
curve contains only finitely many special points. So the both-CM
contribution to the set in Theorem 2.1 is finite.
12.6 CONCLUSION. By 12.1 to 12.4 the mixed points on C are finite, and by
12.5 the both-CM points on C are finite. Together these are exactly the
set in Theorem 2.1, which is therefore finite. That proves Theorem 2.1.
---------------------------------------------------------------------------
13. REFERENCES
---------------------------------------------------------------------------
Chen, Y.-G. On the Siegel-Tatuzawa-Hoffstein theorem. Acta Arith. 130
(2007), 361-367.
Cox, D. A. Primes of the Form x^2 + n y^2: Fermat, Class Field Theory
and Complex Multiplication, 2nd ed. Wiley, 2013.
Daw, C. and Orr, M. Unlikely intersections with E x CM curves in A_2.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), 1705-1745. We use the
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