The E×CM component of Zilber–Pink for curves in A2

1 Introduction

Let \A_2 be the coarse moduli variety of principally polarized abelian surfaces. Its dimension is three. The Zilber–Pink conjecture predicts that a Hodge-generic curve has only finitely many intersections with the union of special subvarieties of codimension at least two. Among these special subvarieties are the E\times\CM curves: one elliptic factor varies, the other has complex multiplication, and one allows all polarized Hecke translates. Here and below, special subvarieties are Shimura special subvarieties, including their Hecke translates. For s\in\A_2(\C), write A_s for a principally polarized abelian surface representing its moduli point. An elliptic curve has CM if its geometric endomorphism algebra is an imaginary quadratic field. In a threefold, a curve and a codimension-two subvariety have negative expected intersection dimension. The conjecture controls the exceptional intersections even as the special subvariety varies. It belongs to the unlikely-intersection framework initiated by Zilber’s atypical-intersection conjectures in semiabelian varieties [24] and formulated for mixed Shimura varieties by Pink [18, Conjecture 1.3].

The isogenies in Theorem 1.1 are isogenies of abelian varieties; they are not required to respect the given polarization. All isogeny degrees, imaginary quadratic fields, and endomorphism orders are included, as is the case of two CM factors. There is no assumption on the boundary of C, on an integral base point, or on reduction at any finite prime. Thus Theorem 1.1 resolves positively the E\times\CM component of Zilber–Pink for Hodge-generic curves in \A_2.

Daw and Orr proved this finiteness statement for curves whose closure meets the zero-dimensional stratum of the Baily–Borel boundary [3, Theorem 1.1]. They also proved that, without this boundary hypothesis, a suitable lower bound for Galois orbits suffices [3, Theorem 1.2]. Their later work proves full Zilber–Pink for curves in \A_2 under the same zero-dimensional-boundary hypothesis [4, Corollary 1.2]. Papas obtains height and finiteness results for Hodge-generic families having an interior fiber isogenous to a square of a non-CM elliptic curve or to a product of one CM and one non-CM elliptic curve. That fiber must have potentially good reduction at every finite place, and the finiteness statement imposes a uniform bound on the number of supersingular places of proximity [13, Theorem 1.5 and Corollary 1.6]. The both-CM case follows from André–Oort for \A_2, proved by Pila and Tsimerman [15, Theorem 1.1] and included in Tsimerman’s general theorem for \A_g [23, Theorem 5.3]. Our task is to supply the orbit bound for the entire mixed locus. We call a point mixed when its abelian surface is isogenous to a product of one CM and one non-CM elliptic curve.

The passage from Galois orbits to finiteness is an instance of the strategy of Pila and Zannier [17]: compare arithmetic lower bounds for Galois orbits with the subpolynomial bounds in the Pila–Wilkie theorem [16]. Daw–Orr establish the required geometric and counting implication for this locus. The arithmetic estimates here use the theory of periods and minimal abelian subvarieties developed by Masser and Wüstholz [10], in the arbitrary-polarization form of Gaudron and Rémond [6]. The degeneration argument uses the integral compactifications of Faltings–Chai and Lan [5, 8], within the reduction and monodromy framework of Grothendieck [7] and Tate’s explicit elliptic uniformization [22].

There are two arithmetic ingredients. First, the product-isogeny description of Daw–Orr [3, Lemmas 2.1 and 3.2] gives a mixed principally polarized surface a canonical graph presentation

where E is non-CM, F is CM, and the pullback polarization is n times the product polarization. At a finite place where E is a Tate curve, the rank-one degeneration length of A is \log|j_E|/n. Combining this reciprocal factor with the height machine on a fixed curve gives

for all sufficiently large n. The argument works both when the curve meets the boundary and when it does not.

Second, we use the gluing data to construct an abelian threefold isogenous to E^2\times F, with an essential period involving both periods of E. A weighted polarization balances their possibly very different lengths. The period theorem of Gaudron and Rémond then produces a factor n^{4/3}, which exceeds the linear dependence on n in (1.1). This construction and the uniform height estimate are the additional inputs beyond the existing unlikely-intersection theorem.

We obtain the following explicit, deliberately coarse form of the arithmetic conclusion. At a mixed point s, let N(s) be the minimum degree of an isogeny from an elliptic product to A_s. The CM factor in a product attaining this minimum is unique up to isomorphism; this will be proved in Lemma 3.1. Write

The proof below uses Siegel’s potentially ineffective class-number bound. Remark 2.3 gives an effective alternative for that input alone; we make no effectivity claim for the constants in Theorem 1.2. The exponent records the positive power needed by the counting theorem; its optimization is not needed here. Section 2 fixes the height and period normalizations and collects the CM estimates. Section 3 proves the canonical gluing description and its reciprocal degeneration formula. Section 4 turns that formula into the height bound on the fixed curve, treating an empty boundary separately. Section 5 constructs and balances the essential period. Section 6 combines the estimates to prove Theorem 1.2, and Section 7 applies the corrected Daw–Orr implication and André–Oort.

2 Heights, periods, and CM elliptic curves

2.1 Conventions

Endomorphism rings and homomorphism groups are geometric unless a ground field is indicated. All heights are absolute logarithmic heights. We write h(j_P) for the usual height of the j-invariant of an elliptic curve P, and \hF(B) for stable Faltings height. Set \hFp(B)=\max\{1,\hF(B)\}. For a line bundle L on a g-dimensional abelian variety, \deg_L B=c_1(L)^g[B]. Thus a principal polarization has degree g! in this convention. The associated period norm is the norm of its positive Hermitian form.

For an elliptic curve with its principal polarization, choose a period basis e,e\tau with \tau in the standard closed fundamental region. Writing y=\Im\tau, one has

The Fourier expansion of j, together with compactness away from the cusp, gives

Constants in these two formulas are absolute. Other implicit constants may depend on a fixed curve, family, and ground field, but never on a varying point or its field of definition.

For a number field k, absolute values at finite places extend the standard absolute values on \Q; they are not renormalized after a field extension. Heights are sums with weights [k_v:\Q_v]/[k:\Q]. Equivalently, the archimedean contribution is the average over all complex embeddings, with conjugate embeddings counted separately.

2.2 Two established height and period estimates

We use product additivity and the isogeny inequality for stable Faltings height:

for an isogeny \phi:B_1\to B_2; see [6, §2.3]. For elliptic curves, \hF is bounded below and

These are stable-height statements, so passing to a semistable extension does not change the heights. For the comparison with j-height, see Silverman [21, Proposition 2.1] and the normalization recorded in [9, Proposition 3.1].

Call a nonzero period \omega of a complex abelian variety essential if it belongs to the Lie algebra of no proper abelian subvariety. For a polarized abelian variety (B,L) of dimension g over a number field k, and an essential period \omega at one complex embedding of k, [6, Theorem 1.2] gives

To explain the field-degree factor, let \delta_0(B_\sigma,L_\sigma) be the least norm of an essential period at the embedding \sigma, with value +\infty if there is none. The essential-minimum form of their theorem bounds

At the chosen embedding, \delta_0(B_\sigma,L_\sigma)\le\|\omega\|_L. Keeping this one nonnegative summand proves (2.5). The estimate allows arbitrary polarization degree.

2.3 CM estimates

3 Canonical gluing and boundary lengths

3.1 The canonical product isogeny

We first identify the product and gluing integer intrinsically, so that the subsequent height bounds use the same complexity as the Galois-orbit theorem. The description below follows [3, Lemmas 2.1 and 3.2]; we include the argument to make the minimality and field of definition explicit.

The Galois action on the endomorphism ring of a CM elliptic curve has image of order at most two: it acts on its imaginary quadratic endomorphism algebra. Thus, when A is defined over a number field k_1, all endomorphisms of F are defined after an extension of k_1 of degree at most two. Defining the graph itself requires no such extension.

3.2 The degeneration input

The new local issue is how the integer n changes the degeneration of the non-CM factor. We first compute the resulting length from semiabelian uniformization, then compare it with the boundary of a fixed integral compactification.

Suppose a principally polarized abelian variety over a complete discretely valued field has split semiabelian reduction. Its uniformization consists of an extension of a good-reduction abelian variety by a split torus, modulo a period lattice Y. If X is the character lattice of the torus, a polarization gives a map Y\to X, an isomorphism for a principal polarization. The degeneration data carry a functorial valuation pairing

We use the extended absolute value in defining this pairing. In toric rank one, its value on generators identified by the principal polarization is a positive number, denoted \ell(A).

To use this reciprocal length in a height estimate on the curve, we need to measure it by a boundary divisor on the moduli space. The following comparison provides this link uniformly at finite places.

We specify the part of integral toroidal compactification theory used in the proof. At full symplectic level m\ge3, choose a smooth projective toroidal compactification of the Siegel moduli scheme over \Z[\zeta_m,1/m], with reduced boundary divisor D_m. The boundary is an effective relative Cartier divisor.

4 A uniform height estimate on a curve

Fix a smooth curve S over a number field K_0, carrying a principally polarized abelian scheme of relative dimension two with full level divisible by 12. Let \bar S be its smooth projective completion. The two level-forgetting maps extend, by properness, to morphisms

Let \Delta be the reduced inverse image of the boundary. Its support is the same at the two levels, because it is characterized by nonzero potential toric rank. It may be empty.

No Hodge-genericity hypothesis is needed for this height estimate. When the boundary is nonempty, a height associated with it controls h(j_E), with a logarithmic isogeny error. We will obtain a reverse bound in which h(j_E) occurs with coefficient O(1/n), allowing absorption for large n. When the boundary is empty, the finite-place estimates and compactness at infinity give the bound directly.

4.1 Uniform local height comparisons

We first make explicit the uniformity in the height machine. An M_{K_0}-bound means constants c_v, zero at all but finitely many finite places of K_0, inherited at places above v by extension of its absolute value. Their weighted sum is bounded independently of any finite extension of K_0.

If \Delta\ne\varnothing, write f_m^*D_m=\Delta+R_m, where R_m is effective. Local heights of effective divisors on a proper variety are bounded below. Functoriality therefore gives

If \Delta=\varnothing, the generic pullback divisor is zero, so instead

These errors are M_{K_0}-bounds. At finite places prime to m, we may use exactly the model local height described in Lemma 3.3.

Here is a model justification that also handles the finitely many bad primes of the curve and of its maps. Over \mathcal O_{K_0}[1/m], take a proper model of \bar S, then the closure of the graph of f_m in its product with the fixed integral toroidal compactification. If needed, pass to a fixed further model on which the relevant divisors are Cartier. Models of the same generic divisor differ by vertical divisors at finitely many primes. An effective vertical Cartier divisor V is bounded above by a fixed multiple of the full special fiber: locally a power of the uniformizer lies in the ideal of V, and quasi-compactness supplies a common power. Taking positive and negative bounds for vertical discrepancies shows that changing models introduces an error bounded in the extended absolute value. This proves the asserted uniformity even when the point and the ramification of its field vary.

Combining (4.1) with Lemmas 3.3 and 3.2, and choosing m\in\{3,4\} prime to the residue characteristic, gives, when \Delta\ne\varnothing,

with fixed c. If \Delta=\varnothing, the two-sided comparison in Lemma 3.3 and (4.2) instead give

The bounds include the potential-good-reduction case, where the boundary value and \log^+|j_E|_v both vanish.

4.2 An archimedean boundary estimate

At a complex point, let \rho(A) be the length of the shortest nonzero period of A for its principal polarization.

4.3 Proof of Proposition 4.1

Suppose first that \Delta=\varnothing. At every embedding of K_0, the image of \bar S is compact in the interior moduli space. The shortest period length has a positive lower bound on this image. Lemma 4.3 therefore gives y_E\ll n at every embedding of a field of definition of the point. Equations (2.2) and (4.4), summed with the normalized local weights, show

This proves the proposition in this case.

Now let \Delta\ne\varnothing. Choose h_\Delta shifted to be nonnegative. It is an ample height because \Delta is a nonzero effective divisor on a projective curve. Lemmas 4.2 and 4.3, together with (2.7), give at the archimedean places

Use (2.2) for y_E, add (4.3), and sum. The M_{K_0}-bounds give a constant independent of the field. We obtain

for a fixed B>0.

On the other hand, (2.3) gives \hF(E)+\hF(F)\le\hF(A)+\log n. The lower bound for elliptic Faltings height, (2.4), and Lemma 2.1 imply

for fixed A_0>0. Combining (4.6) and (4.7), the coefficient of h(j_E) on the right is A_0B/n. For n\ge2A_0B it can be absorbed. Since \log n\ll n, this gives h(j_E)\ll n(1+\sqrt D), as claimed. □

5 An essential period on a threefold

We now extract a stronger power of the gluing index by repeating the non-CM factor. This lets one period vector contain both independent periods of E, so no nonzero integer row on E^2 can annihilate its E-coordinates. Repeated elliptic factors and polarizations adapted to reduced period bases are used in [6, §§7.2–7.3]; here the gluing data produce a threefold isogenous to E^2\times F. All constructions in this section are over a number field k 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 \Lambda_E, and a first reduced period f of F. Use b,y_F from Lemma 2.2, 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 approximately y^2 makes their contributions comparable. Weighting F by approximately y then makes the cube root of the product polarization degree grow at the same rate as the squared norm. The remaining gain is the power n^{4/3} contributed by the quotient.

6 The Galois orbit bound

We apply the preceding estimates to the curve of Theorem 1.2. Pass to a component of a full-level cover with level divisible by 12, and normalize it. After choosing a number field K_0, enlarged once and for all, this gives a smooth curve S/K_0 with a family as in Section 4, and a map S\to C of fixed bounded degree over a dense open. Further fixed covers or restrictions used to describe the family have the same property. Only finitely many points of C are omitted. This passage is permitted for a singular C as well.

If s belongs to that dense open, choose a lift t with

Such a lift exists because the finite fiber has bounded degree, and K_0/K is fixed. By Lemma 3.1, E,F,\theta are defined over the field of the fiber. After an extension of degree at most two, all CM endomorphisms are defined too. For the resulting field k, put d=[k:\Q]; then

It remains to bound \max\{n^2,D\} by a fixed power of d.

For n\ge n_0, Lemma 2.2 and Proposition 4.1 give

Let (B,L) be the threefold from Proposition 5.1. Its height satisfies

To bound its polarization degree, use (2.2) at the chosen embedding. A single archimedean contribution to absolute height is at most d times the total, so

It follows that \log(6n^4uv)\ll1+\log n+\log d\ll nd^2. Thus (5.1) and (6.2) give

Dividing by n and cubing proves

For the remaining n<n_0, the same final bound follows from D\ll d^4 by increasing the constant.

By Lemma 3.1, \cplx(s)=\max\{n^2,D\}. Equations (6.1) and (6.3) therefore imply

The degree on the left is the cardinality of the Galois orbit. The finitely many omitted mixed points can be included by reducing the positive constant. This proves Theorem 1.2. □

7 Finiteness on the original curve

We recall exactly the unlikely-intersection implication being used. Daw–Orr [3, Theorem 1.2] proves that a Hodge-generic curve in \A_2 has finitely many mixed E\times\CM points if their Galois orbits satisfy a power lower bound in the relevant special curve complexity. Their alternative complexity is the maximum of the minimum product-isogeny degree and the endomorphism-ring discriminant of the CM factor in a product attaining that degree; it is polynomially comparable with their original complexity [3, §3, especially Lemmas 3.1 and 3.6]. It is exactly \cplx(s) by Lemma 3.1. Theorem 1.2 supplies the required bound. No boundary hypothesis is present in this implication.¹

The mixed points occurring in the Daw–Orr formulation are algebraic in our setting. Indeed, a mixed complex point lies on an E\times\CM special curve defined over \Qbar: start with the diagonal product locus with a fixed CM elliptic coordinate and use the polarized isogeny of Lemma 3.1 to pass through a Hecke correspondence. Such a special curve meets C in a zero-dimensional set, because C is Hodge generic. These intersection points are algebraic. The elliptic subvarieties and their homomorphisms are then available over \Qbar, by invariance of homomorphisms of abelian varieties under algebraically closed extension. Thus the complex and algebraic mixed loci required here coincide.

The same orbit bound supplies the finitely-generated-field version of Daw–Orr’s hypothesis for this curve over \Qbar. Given a finitely generated field of definition L\subset\C, let K be a number field of definition of C, set M=LK, and put K'=M\cap\Qbar. Then K' is a number field containing K, and M/K' is regular. For an algebraic point s, linear disjointness shows that its \operatorname{Aut}(\C/M)-orbit has cardinality [K'(s):K']. Theorem 1.2 applies over K'. Since \operatorname{Aut}(\C/M) is a subgroup of \operatorname{Aut}(\C/L), the required lower bound holds for the latter orbit too.

We conclude that C has finitely many mixed points. If both elliptic factors have CM, their product, and hence every isogenous abelian surface, is of CM type. Its moduli point is special. André–Oort for \A_2 gives only finitely many special points on a Hodge-generic curve. Together these two conclusions prove Theorem 1.1 on C itself. □