INDEPENDENCE OF POINTS ON ELLIPTIC CURVES ARISING FROM SPECIAL POINTS ON MODULAR AND SHIMURA CURVES, II: LOCAL

RESULTS

ALEXANDRU BUIUM AND BJORN POONEN

Abstract. In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite rank subgroups, and to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura curve analogues of these results.

1. Introduction

Let N > 3. Let S be the modular curve X_{1}(N) over \overline{Q}. Let CM ⊆ S(\overline{Q}) be the set of CM-points on S. (See Section 2 for definitions.) Let A be an elliptic curve over \overline{Q}. Given a morphism S → A, we may map the CM-points on S to points on A, and ask what relations exist among them in the group law on A. More generally, we may consider a modular-elliptic correspondence, a pair of non-constant morphisms S ⟵^{Π} X ⟶^{Φ} A of smooth connected projective curves over \overline{Q}, where S and A are as above. On the one hand, it is easy to construct some relations by using Hecke correspondences: see (A.4). On the other hand, the following special case of Theorem 2.1 of [8] says that not too many relations exist:

Theorem 1.1. Let S ⟵^{Π} X ⟶^{Φ} A be a modular-elliptic correspondence and let Γ ≤ A(\overline{Q}) be a finite rank subgroup. Then Φ(Π^{−1}(CM)) ∩ Γ is finite.

(Recall Γ is said to be of finite rank if the quantity rank(Γ) := dim_{Q}(Γ ⊗ Q) is finite.) Theorem 2.5 of [8] implies an analogous result when S is a Shimura curve and Π is the identity.

The aim of this paper is to prove local analogues of these results in which, roughly speaking, the field \overline{Q} is replaced by the completion R := Ẑ_{p}^{ur} of the maximal unramified extension of the ring Z_{p} of p-adic integers, and the set CM is replaced by either the set CL of canonical lift points or by a fixed (partial) isogeny class. The new results represent an improvement over those in [8] in that they come with effective bounds and are valid for certain groups Γ of infinite rank.

For historical background see Section 1.2 of [8], which comments on related results in [12, 24, 28, 31, 35, 45]. Although the present article is intended as a sequel to [8], it is logically independent of [8].

Date: October 20, 2008.

2000 Mathematics Subject Classification. Primary 11G18; Secondary 14G20.

Key words and phrases. modular curve, Shimura curve, CM point, canonical lift, reciprocity law.

Our methods are quite different from those used in [8, 12, 24, 28, 31, 35, 45]. Indeed, our local results will be proved using the theory of arithmetic differential equations in the sense of [7]: see Section 3.9.

1.1. Main theorems. First we introduce the following local analogue of rank, in order to treat some infinite-rank groups as if they were of finite rank.

Definition 1.2. For any abelian group G define G_{p-div} := G_{tors} + pG. For any subgroup Γ ≤ G define

rank_{p}^{G}(Γ) := dim_{F_{p}} (\frac{Γ}{Γ ∩ G_{p-div}} ) .

Then

rank_{p}^{G}(Γ) ≤ dim_{Q}(Γ ⊗ Q) =: rank Γ,
rank_{p}^{G}(Γ) ≤ dim_{F_{p}}(Γ ⊗_{Z} F_{p}).

Assume that we are given a modular-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A. For each sufficiently large prime p ∈ Z one can choose a model of this correspondence over R := Ẑ_{p}^{ur}:

see Section 3. We obtain maps S(R) ⟵^{Π} X(R) ⟶^{Φ} A(R). Let CL ⊂ S(R) be the set of CL-points (canonical lift points); see Section 3.3 for more on the definition of CL. In an appropriate sense, CL is a subset of CM: see Theorem 4.4.

Theorem 1.3 (Finiteness for CL points in a subgroup). Suppose that S ⟵^{Π} X ⟶^{Φ} A is a modular-elliptic or Shimura-elliptic correspondence (see Section 2 for definitions) and assume that p is a sufficiently large good prime in the sense of Definition 3.3. Then there exists a constant c depending on p such that for any subgroup Γ ≤ A(R) with r := rank_{p}^{A(R)}(Γ) < ∞, the set Φ(Π^{−1}(CL)) ∩ Γ is finite of cardinality at most cp^{r}.

Remark 1.4. Corollary 3.19 makes c explicit in the case where Π is the identity and Φ is a modular parametrization in the sense of Definition 2.3.

Remark 1.5. There are interesting examples of subgroups Γ ≤ A(R) with rank_{p}^{A(R)}(Γ) < ∞ and rank(Γ) = ∞: indeed, if Γ := Γ_{0} + pA(R), where Γ_{0} ≤ A(R) and rank(Γ_{0}) < ∞, then Γ is such an example; see Remark 3.11 for more on this.

If S is a modular curve and Σ is a set of prime numbers, define the Σ-isogeny class of Q in S(R) as the set of all points in S(R) corresponding to elliptic curves that admit an isogeny u to E such that all the prime divisors of deg(u) are in Σ; there is a similar definition in the Shimura curve case: see Section 3.7 for details. Also, if S is a modular curve, and Q ∈ S(R) is an ordinary point, i.e., a point corresponding to an elliptic curve E with good ordinary reduction \overline{E}, then let K_{Q} := End(\overline{E}) ⊗ Q; for the similar definition in the Shimura curve case, see Section 3.4.

Theorem 1.6 (Finiteness of the intersection of an isogeny class with a subgroup). Assume that S ⟵^{Π} X ⟶^{Φ} A is a modular-elliptic or Shimura-elliptic correspondence and that p is a sufficiently large good prime. Let Q ∈ S(R) be an ordinary point. Let Σ be the set of all rational primes that are inert in the imaginary quadratic field K_{Q}. Let C be the Σ-isogeny class of Q in S(R). Then there exists a constant c such that for any subgroup Γ ≤ A(R) with r := rank_{p}^{A(R)}(Γ) < ∞ the set Φ(Π^{−1}(C)) ∩ Γ is finite of cardinality at most cp^{r}.

Theorems 1.1, 1.3, and 1.6 suggest the following “global” conjecture:

Conjecture 1.7. Let S ⟵^{Π} X ⟶^{Φ} A be a modular-elliptic or Shimura-elliptic correspondence. Let Γ ≤ A(\overline{Q}) be a finite rank subgroup. Let C ⊂ S(\overline{Q}) be an isogeny class. Then the set Φ(Π^{−1}(C)) ∩ Γ is finite.

1.2. Reciprocity functions. Our local results for CL points are proved via “reciprocity theorems” (e.g., Theorem 3.5), which transform relations between certain CL-points in A into additive relations between values of a certain “reciprocity function”. More precisely, one part of Theorem 3.5 (with Remark 3.7 for terminology) shows that given a modularelliptic correspondence S ⟵^{Π} X ⟶^{Φ} A and a model of X over R there exist an affine dense open subscheme X^{†} of this model and a p-adic formal function Φ^{†} on X^{†} with nonconstant reduction such that for any divisor ∑ m_{i}P_{i} supported on Π^{−1}(CL) ∩ X^{†}(R), we have ∑ m_{i}Φ(P_{i}) ∈ A(R)_{tors} if and only if ∑ m_{i}Φ^{†}(P_{i}) = 0 ∈ R.

Some reciprocity results have analogues for (local) isogeny classes: see Sections 3.7 and 3.8. On the other hand, Theorems A.2, A.1, and A.10 show that there is no reciprocity in the

global setting.

1.3. Structure of the paper. We review basic definitions in Section 2. Section 3 states all our local results beyond those already in this introduction, and Section 4 proves them. Section 4 also reviews the necessary background from the theory of arithmetic differential equations. The non-existence of global reciprocity functions is relegated to an appendix.

Remark 1.8. The proofs of the modular and Shimura cases are parallel and share some common tools, but they are logically independent in the sense that it is not necessary to follow both cases to understand only one. A similar comment applies to results for CL points versus isogeny classes.

The following leitfaden may help the reader seeking a quick path through the proof of the modular case of Theorem 1.3 (the result for CL points). Theorem 1.3 follows from Theorem 3.5 and its immediate Corollary 3.8. The proof of Theorem 3.5 is sketched in Section 3.9. A reader accepting parts (4) and (5) of Lemma 4.7, the isomorphism (4.48), and formula (4.15) can go directly to the first paragraphs of Section 4.9 for a complete proof of Theorem 3.5 in the modular case.

Acknowledgments. While writing this paper, the authors were partially supported by NSF grants: A.B. by DMS-0552314, and B.P. by DMS-0301280 and DMS-0841321. We are indebted to M. Kim, J. H. Silverman, and J. F. Voloch for their remarks and suggestions. The paper also benefited greatly from the advice of a referee.

2. Basic definitions

2.1. Modular curves. Let N ∈ Z satisfy N > 3. Let X_{1}(N) over \overline{Q} be the complete modular curve attached to the group Γ_{1}(N). If Y_{1}(N) ⊂ X_{1}(N) is the non-cuspidal locus then Y_{1}(N)(\overline{Q}) is in bijection with the set of isomorphism classes of pairs (E, α) where E is an elliptic curve over \overline{Q} and α: Z/NZ ↪ E(\overline{Q}) is an injection.

Definition 2.1. A CM-point on the curve S := X_{1}(N) is a point in Y_{1}(N)(\overline{Q}) represented by a pair (E, α) such that E has complex multiplication, i.e., End(E) ≠ Z. Let CM ⊂ S(\overline{Q}) be the set of CM-points on S.

Definition 2.2. A modular-elliptic correspondence is a pair of non-constant morphisms of smooth connected projective curves over \overline{Q}, S ⟵^{Π} X ⟶^{Φ} A, where S = X_{1}(N) and A is an elliptic curve. From now on, we normalize Φ by fixing x_{∞} ∈ X(\overline{Q}) such that Π(x_{∞}) = ∞ and requiring Φ(x_{∞}) = 0. Call Φ(Π^{−1}(CM)) ⊂ A(\overline{Q}) the set of CM-points on A.

Definition 2.3. Let f = ∑ a_{n}q^{n} be a newform. (Unless otherwise specified, newforms in this paper are of weight 2, on Γ_{0}(N), and normalized (a_{1} = 1), with Fourier coefficients in Z.) (For terminology on modular forms we refer to [13].) The Eichler-Shimura construction [13] yields a Q-morphism from X_{0}(N) to an elliptic curve A_{f}. By a modular parametrization attached to f we mean a composition X_{1}(N) → X_{0}(N) → A_{f} → A where X_{1}(N) → X_{0}(N) is the usual map and A_{f} → A is any isogeny of elliptic curves over Q. A modular-elliptic correspondence is said to arise from a modular parametrization if it is of the form S ⟵^{Π} X ⟶^{Φ} A where S = X = X_{1}(N), Π = Id, and Φ is a modular parametrization.

Remark 2.4. By work of Wiles and others [47, 44, 2], together with the Isogeny Theorem of Faltings [16], any elliptic curve A over Q has a modular parametrization.

Definition 2.5. The isogeny class C of a non-cusp Q ∈ S(\overline{Q}) is the set of points in S(\overline{Q}) such that the corresponding elliptic curve admits an isogeny to the elliptic curve corresponding to Q. (The isogeny is not required to respect the points of order N.)

2.2. Shimura curves. Let D be a non-split indefinite quaternion algebra over Q. Fix a maximal order O_{D} once and for all. Let X^{D}(U) be the Shimura curve attached to the pair (D, U), where U is a sufficiently small compact subgroup of (O_{D} ⊗ (\underleftarrow{lim} Z/mZ))^{×} such that X^{D}(U) is connected: see [9, 48].

Definition 2.6. A fake elliptic curve^{1} is a pair (E, i) consisting of an abelian surface E over \overline{Q} and an embedding i: O_{D} → End(E).

The set X^{D}(U)(\overline{Q}) is in bijection with the set of isomorphism classes of fake elliptic curves equipped with a level U structure in the sense of [9, 48].

Definition 2.7. The classification of endomorphism algebras [30, p. 202] shows that for any fake elliptic curve (E, i), the algebra (End E)⊗Q is isomorphic to either D or D⊗K ≃ M_{2}(K) for some imaginary quadratic field K embeddable in D. In the latter case, (E, i) is called CM; then E is isogenous to the square of an elliptic curve with CM by an order in K. A CM-point of S(\overline{Q}) is a point whose associated (E, i) is CM. Let CM ⊂ S(\overline{Q}) be the set of CM-points on S.

Definition 2.8. A Shimura-elliptic correspondence is a pair of non-constant morphisms of smooth connected projective curves over \overline{Q}, S ⟵^{Π} X ⟶^{Φ} A, where S is a Shimura curve as above and A is an elliptic curve. Call Φ(Π^{−1}(CM)) ⊂ A(\overline{Q}) the set of CM-points on A.

Definition 2.9. For any Q ∈ S(\overline{Q}), represented by a fake elliptic curve (E, i) with level U-structure, the isogeny class C of Q in S(\overline{Q}) consists of all points in S(\overline{Q}) represented by fake elliptic curves (E′, i′) with level U-structure such that there is an isogeny E → E′ compatible with the O_{D}-action (but not necessarily compatible with the level U-structures).

1In the literature this is sometimes called a “false elliptic curve”.

3. Detailed exposition of the results

3.1. Review of Witt rings. Fix a prime p. Let Z_{p} be the ring of p-adic integers. Let Z_{p}^{ur} be the maximal unramified extension of Z_{p}. Let R := Ẑ_{p}^{ur} be the completion of Z_{p}^{ur}. We set k = R/pR and K := R[1/p]. Thus k ≃ \overline{F}_{p}, and R is the Witt ring W (k). Let Fr: k → k be the automorphism Fr(x) := x^{p}, and let ϕ: R → R be the unique automorphism lifting Fr.

We will use the notion of a canonical lift (CL) abelian scheme over R: see Section 4.1 for the definition.

3.2. Hecke correspondences. For any prime l let Y_{1}(N, l) be the affine curve over \overline{Q} parametrizing triples (E, α, H) in which (E, α), with α: Z/NZ ↪ E(\overline{Q}), represents a point in Y_{1}(N) and H ≤ E(\overline{Q}) is an order-l subgroup intersecting α(Z/NZ) trivially: see [11, p. 207]. Define degeneracy maps σ_{1}, σ_{2} : Y_{1}(N, l) → Y_{1}(N) by σ_{1}(E, α, H) := (E, α) and σ_{2}(E, α, H) := (E/H, u ∘ α), where u: E → E/H is the quotient map.

Let X_{1}(N, l) be the smooth projective model of Y_{1}(N, l). The σ_{i} extend to σ_{i}: X_{1}(N, l) → X_{1}(N). Define the Hecke operator T (l)_{∗} on Div(X_{1}(N)(\overline{Q})) by T (l)_{∗}D := σ_{2∗}σ_{1}^{∗}D. For P ∈ X_{1}(N)(\overline{Q}) write T (l)_{∗}P =: ∑_{i} P_{i}^{(l)}; the sum involves l + 1 or l terms according as l ∤ N or l | N. If in addition f = ∑ a_{n}q^{n} ∈ Z[[q]] is a newform, then the divisor ∑_{i} P_{i}^{(l)} − a_{l}P will be called a Hecke divisor.

3.3. Conventions on modular-elliptic correspondences. The Z[1/N]-scheme Y_{1}(N) represents the functor taking a Z[1/N]-algebra B to the set of isomorphism classes of pairs (E, α) where E is an elliptic curve over B and α: (Z/NZ)_{B} → E is a closed immersion of group schemes. For each P ∈ Y_{1}(N)(B), let (E_{P} , α_{P}) be a pair in the corresponding isomorphism class. The Z[1/N]-scheme S = X_{1}(N) is the Deligne-Rapoport compactification: see [13, pp. 78–81]. The base extension of S to C will also be denoted S. The cusp ∞ on X_{1}(N) is defined over Q(ζ_{N}), where ζ_{N} is a primitive N^{th} root of 1.

Remark 3.1. Some of the references we cite use a modular curve parametrizing elliptic curves with an embedding of μ_{N} instead of Z/NZ, but the two theories are isomorphic provided we work over Z[1/N, ζ_{N}]-algebras.

Assume that we are given a modular-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A with S = X_{1}(N). We may assume that A comes from a model over O_{F_{0}}[1/Nm], and that X, S, Π, Φ come from models over O_{F} [1/Nm], where F_{0} ⊆ F are number fields, and O_{F_{0}} and O_{F} are their rings of integers, and m ∈ Z_{>0}. Then x_{∞} = Π(∞) has a model over O_{F_{1}}[1/Nm], where F_{1} is a number field containing F (ζ_{N}).

If p is large enough to be unramified in F_{1}, then we fix once and for all an embedding of F_{1} into K; then we obtain an embedding O_{F_{1}}[1/Nm] ⊂ R. A point P ∈ S(R) is called ordinary (respectively, a CL-point) if P ∈ Y_{1}(N)(R) and E_{P} has ordinary reduction \overline{E}_{P} (respectively, E_{P} is CL). If P ∈ S(R) is ordinary let K_{P} be the imaginary quadratic field End(\overline{E}_{P}) ⊗ Q. Finally let CL be the set of all CL-points of S(R). Call Φ(Π^{−1}(CL)) ⊂ A(R) the set of CL-points of A.

3.4. Conventions on Shimura-elliptic correspondences. Now suppose instead that S is a Shimura curve X^{D}(U), where D and U satisfy the conditions in [9]; then for some m ∈ Z_{>0} the Shimura curve S = X^{D}(U) is a Z[1/m]-scheme with geometrically integral fibers, such that for any Z[1/m]-algebra B the set S(B) is in bijection with the set of isomorphism classes of triples (E, i, α) where (E, i) is a fake elliptic curve over B (i.e. E/B is an abelian scheme of relative dimension 2 and i: O_{D} → End(E/B) is an injective ring homomorphism) and α is a level U structure.

Assume that we are given a Shimura-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A. With notation as in Section 3.3, Replacing m by a multiple if necessary, we may assume that A comes from a model over O_{F_{0}}[1/m] and that X, S, Π, Φ come from models over O_{F} [1/m], where F_{0} ⊆ F are number fields. Assuming that p is suitably large, we again fix an embedding F ⊂ K hence we have an embedding O_{F} [1/m] ⊆ R. A point P ∈ S(R) is called ordinary (respectively a CL-point) if P corresponds to a triple (E_{P} , i_{P} , α_{P}) where E_{P} has ordinary reduction \overline{E}_{P} (respectively E_{P} is CL). If P ∈ S(R) is ordinary, let K_{P} be the imaginary quadratic field End(\overline{E}_{P} , \overline{i}_{P}) ⊗ Q. Finally CL is the set of all CL-points of S(R). Call Φ(Π^{−1}(CL)) ⊂ A(R) the set of CL-points of A.

3.5. Reciprocity functions for CL points.

Definition 3.2. Let p be a prime number. Let F_{0} be a number field, and let v be a degree-1 place lying above p. Call v anomalous for an elliptic curve A over F_{0} if the trace a_{v} of the p-power Frobenius on the reduction A mod v satisfies a_{v} ≡ 1 (mod p). (See [27, p. 186].)

Let notation be as in Section 3.3 or Section 3.4.

Definition 3.3. A rational prime p is good (for our correspondence) if p splits completely in F_{0}, the elliptic curve A has good reduction at all primes v|p, and in the Shimura-elliptic case each v|p is not anomalous for A.

Remark 3.4. The Chebotarev density theorem easily implies that there are infinitely many good primes.

Let p be sufficiently large and set X_{R} := X ⊗ R. (More generally, throughout this paper the subscript R always means “base extension to R” and we use the same convention for any other ring in place of R. In particular, if p is a good prime, A_{R} comes from an elliptic curve A_{Z_{p}} over Z_{p} and we let a_{p} be the trace of the p-power Frobenius on A_{F_{p}}.) Let X̄ := X_{k} = X ⊗ k. For any P ∈ X(R), let P̄ denote the image of P in X̄(k). (More generally, throughout this paper, when we are dealing with a situation that is “localized at p”, an upper bar always means “reduction mod p”.) Let X̂_{R} the p-adic completion of X_{R} viewed as a formal scheme over R. (More generally, throughout this paper, an upper ˆ will denote “p-adic completion”.) If X^{†} ⊂ X_{R} is an affine dense open subscheme then any global function Φ^{†} ∈ O(X̂^{†}) = O(X^{†})ˆ defines a map Φ^{†}: X^{†}(R) → R. The reduction \overline{Φ^{†}} ∈ O(X̄^{†}) induces a regular map \overline{Φ^{†}}: X̄^{†}(k) → k.

Recall the group A(R)_{p-div} := A(R)_{tors} + pA(R).

Theorem 3.5 (Reciprocity functions for CL points). Assume that S ⟵^{Π} X ⟶^{Φ} A is a modular-elliptic or a Shimura-elliptic correspondence and that p is a sufficiently large good prime. Then there exist an affine dense open subscheme X^{†} ⊂ X_{R} and a function Φ^{†} ∈ O(X̂^{†}) with non-constant reduction \overline{Φ^{†}} ∈ O(X̄^{†}) \ k, such that for any P_{1}, …, P_{n} ∈ Π^{−1}(CL) ∩ X^{†}(R) and any m_{1}, …, m_{n} ∈ Z we have

∑_{i=1}^{n} m_{i}Φ(P_{i}) ∈ A(R)_{tors} ⟺ ∑_{i=1}^{n} m_{i}Φ^{†}(P_{i}) = 0 ∈ R,
∑_{i=1}^{n} m_{i}Φ(P_{i}) ∈ A(R)_{p-div} ⟺ ∑_{i=1}^{n} m_{i}\overline{Φ^{†}}(P̄_{i}) = 0 ∈ k.

Theorem 3.5 will be proved in Section 4. It is useful to compare Theorem 3.5 to Theorems A.1 and A.10.

Remark 3.6. As the proof of Theorem 3.5 will show, the functions Φ^{†} will be functorially associated (in an obvious sense) to tuples (X, S, A, Π, Φ, ω_{A}), where ω_{A} is a nonzero global 1-form on A defined over F_{0}.

Remark 3.7. Let C = Π^{−1}(CL) ∩ X^{†}(R) and let Div(C) be the free abelian group generated by C. Then one can consider the maps Φ_{∗}: Div(C) → A(R)/A(R)_{tors} and Φ_{∗}^{†}: Div(C) → R naturally induced by Φ and Φ^{†} by additivity. Then the first equivalence in Theorem 3.5 says that Ker(Φ_{∗}) = Ker(Φ_{∗}^{†}). A similar description can be given for the second equivalence. There is a formal similarity between such a formulation of Theorem 3.5 and the way classical reciprocity laws are formulated in number theory and algebraic geometry. Indeed, in classical reciprocity laws one is usually presented with maps Φ : C → G and Φ^{†} : C → G^{†} from a set C of places of a global field to two groups G and G^{†} (typically a Galois group and a class group), and one claims the equality of the kernels of the induced maps Φ_{∗} : Div(C) → G and Φ_{∗}^{†} : Div(C) → G^{†}.

Let us discuss some consequences of Theorem 3.5.

Corollary 3.8. In the notation of Definition 1.2 and Theorem 3.5, we have

rank (∑_{i=1}^{n} Z · Φ(P_{i}))  = rank (∑_{i=1}^{n} Z · Φ^{†}(P_{i}))
rank_{p}^{A(R)} (∑_{i=1}^{n} Z · Φ(P_{i})) = dim_{F_{p}} (∑_{i=1}^{n} F_{p} · \overline{Φ^{†}}(P̄_{i})) .

Proof of Theorem 1.3. By Corollary 3.8, the F_{p}-span of

\overline{Φ^{†}}(\overline{Φ^{−1}(Γ) ∩ Π^{−1}(CL) ∩ X^{†}(R)})

has dimension ≤ r over F_{p}. So

#\overline{Φ^{−1}(Γ) ∩ Π^{−1}(CL) ∩ X^{†}(R)} ≤ p^{r} deg(\overline{Φ^{†}}).

Now CL elliptic curves over R are uniquely determined, up to isomorphism, by their reduction mod p: see Theorem 4.3. Similarly, by loc. cit., if (E_{1}, i_{1}) and (E_{2}, i_{2}) are two fake elliptic curves such that E_{1}, E_{2} are CL and (Ē_{1}, ī_{1}) ≃ (Ē_{2}, ī_{2}) then (E_{1}, i_{1}) ≃ (E_{2}, i_{2}). Thus

Φ^{−1}(Γ) ∩ Π^{−1}(CL) ∩ X^{†}(R)

has at most p^{r} deg(\overline{Φ^{†}})·d_{1}d_{2} elements, where d_{1} := deg Π and d_{2} is the number of level Γ_{1}(N) structures (respectively, level U structures) on a given elliptic (respectively, fake elliptic) curve. Also, # (Π^{−1}(CL) \ X^{†}(R)) ≤ d_{1}d_{2}d_{3}, where d_{3} = # (Π^{−1}(\overline{S}^{ord}(k)) \ X̄^{†}(k)), where the ord superscript indicaes the ordinary locus. So

(3.9)  #Φ(Π^{−1}(CL)) ∩ Γ ≤ #Φ^{−1}(Γ) ∩ Π^{−1}(CL) ≤ (p^{r} deg(\overline{Φ^{†}}) + d_{3})d_{1}d_{2},

which is at most cp^{r}, where c := deg(\overline{Φ^{†}}) + d_{1}d_{2}d_{3}. □

Corollary 3.19 will make the bound in (3.9) explicit in the case where S = X = X_{1}(N), Π = Id, and Φ is a modular parametrization.

Remark 3.10. Let M be the algebraic closure of F in K. Then the study of pA(M) is analogous to the study of Wieferich places in [43] and [46]: indeed, for a ∈ Z not divisible by p, the classical Wieferich condition a^{p} ≡ a (mod p^{2}) is equivalent to a ∈ M^{×p}, and M^{×p} is the analogue of pA(M) for the multiplicative group G_{m}.

Remark 3.11. Let Γ_{0} be a finite-rank subgroup of A(M), and let Γ := Γ_{0} + pA(M). Then rank_{p}^{A(R)}(Γ) < ∞, so Theorem 1.3 applies to Γ.

On the other hand, we claim that rank(Γ) = ∞. This follows from the following statement: If L is the compositum in \overline{F} of all quadratic extensions of F that are unramified at all primes above p, then A(L) is of infinite rank. To prove this, choose a Weierstrass equation y^{2} = f(x) for A, where f(x) is a monic cubic polynomial with coefficients in the ring of integers O_{F} of F. Consider points with x-coordinate x_{n} = 1/p^{4} + n for n ∈ O_{F}. Then F (\sqrt{f(x_{n})}) is unramified at p since the equation p^{12}f(x_{n}) ≡ 1 (mod p^{4}) implies by Hensel’s lemma that p^{12}f(x_{n}) is a square in the completion of F at any prime above p. Thus we get a collection of points in A(L). We may inductively define a sequence of n_{i} ∈ O_{F} such that each F (\sqrt{f(x_{n_{i}})}) is ramified at a prime of F not ramifying in the field generated by the previous square roots, by choosing n_{i} so that 1/p^{4} + n_{i} has valuation 1 at some prime of F splitting completely in the splitting field of f. By choosing the n_{i} sufficiently large, we may assume that the corresponding points P_{i} ∈ A(L) have large height and hence are non-torsion. Now we claim that the Galois action forces P_{1}, …, P_{m} to be Z-independent in A(L). Indeed, if there were a relation a_{1}P_{1} + ⋯ + a_{m}P_{m} = 0 then we could apply a Galois automorphism fixing all the P_{i} but P_{1} to obtain −a_{1}P_{1} + a_{2}P_{2} + ⋯ + a_{m}P_{m} = 0, and subtracting would show that 2a_{1}P_{1} = 0, but P_{1} is non-torsion, so a_{1} = 0; similarly all a_{i} would be 0. Since m can be made arbitrarily large, A(L) has infinite rank.

3.6. Refinement of results on CL points for modular parametrizations. Theorem 3.17 below is a refinement of Theorem 3.5 in the special case of a modular-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A arising from a modular parametrization attached to a newform f = ∑ a_{n}q^{n}; recall that S = X = X_{1}(N), Π = Id, and we always assume f of weight 2, on Γ_{0}(N), normalized, with rational Fourier coefficients. In this case we may (and will) take F = F_{0} = Q. Recall that a_{1} = 1, that a_{n} ∈ Z for n ≥ 1, and that for sufficiently large p, the coefficient a_{p} equals the trace of Frobenius on A_{F_{p}}. One can ask if in this case the function Φ^{†} also has a description in terms of eigenforms. This is indeed the case, as we shall explain below. Consider the series

(3.12)  f^{(−1)}(q) := ∑_{(n,p)=1} \frac{a_{n}}{n} q^{n} ∈ Z_{p}[[q]].

The series f^{(−1)}(q) is called f|R_{−1} in [38, p. 211]. Assume that p ≫ 0 and that A_{R} has ordinary reduction. Then a_{p} ≢ 0 (mod p). Let up ∈ Z_{p}^{×} be the unique root in pZ_{p} of the equation x^{2} − a_{p}x + p = 0; thus ā_{p}ū = 1. Let V : Z_{p}[[q]] → Z_{p}[[q]] be the operator V (∑ c_{n}q^{n}) = ∑ c_{n}q^{np}. Define

(3.13)  f_{[u]}^{(−1)}(q) := (∑_{i=0}^{∞} u^{i}V^{i}) f^{(−1)}(q) = ∑_{i≥0} ∑_{(n,p)=1} u^{i} \frac{a_{n}}{n} q^{np^{i}} ∈ Z_{p}[[q]].

Then

(3.14)  − (\overline{f_{[u]}^{(−1)}(q)})^{p} + ā_{p}\overline{f_{[u]}^{(−1)}(q)} = ā_{p}\overline{f^{(−1)}(q)},

in F_{p}[[q]], where the bars denote reduction modulo p, as usual. The series \overline{f^{(−1)}(q)} has a nice interpretation in terms of modular forms mod p. Indeed, recall from [17, pp. 451, 458] that if M_{m} is the k-linear space of modular forms over k on Γ_{1}(N) of weight m then there is an injective q-expansion map M_{m} → k[[q]] and a Serre operator θ: M_{m} → M_{m+p+1} that on q-expansions acts as q \frac{d}{dq}. Let Ē_{p−1} ∈ M_{p−1} be the reduction mod p of the modular form E_{p−1} over Z_{(p)} whose q-expansion in Z_{(p)}[[q]] is the normalized Eisenstein series of weight p − 1; hence Ē_{p−1} is the Hasse invariant and has q-expansion 1 in F_{p}[[q]].

Define the affine curve

\overline{X_{1}(N)}^{ord} := \overline{X_{1}(N)} \ \{zero locus of Ē_{p−1}\}. = \overline{Y_{1}(N)}^{ord} ∪ \{cusps\}

where \overline{Y_{1}(N)}^{ord} is the open set of points in \overline{Y_{1}(N)} represented by ordinary elliptic curves.

If α ∈ M_{m+w}, and β ∈ M_{m} is nonzero, call α/β a weight-w quotient of modular forms over k. A weight-0 quotient of modular forms is a rational function on \overline{X_{1}(N)}. In particular, θ^{p−2}f̄, Ē_{p−1}^{p} ∈ M_{p^{2}−p}, and

(3.15)  f̄^{(−1)} := (θ^{p−2}f̄)/Ē_{p−1}^{p}

is a regular function on \overline{X_{1}(N)}^{ord}. Let g ↦ g_{∞} be the natural q-expansion map k(\overline{X_{1}(N)}) → k((q)). The corresponding point in \overline{X_{1}(N)}(k((q))) will be called the Fourier k((q))-point. Then f̄_{∞}^{(−1)} = \overline{f^{(−1)}(q)}. For primes l ≠ p, define the Hecke operator T (l): k[[q]] → k[[q]] by T (l)(∑ c_{n}q^{n}) = ∑ c_{ln}q^{n} + ϵ(l)l^{−1} ∑ c_{n}q^{ln}, where ϵ(l) = 0 or 1 according as l divides N or not. Define the U-operator U: k[[q]] → k[[q]] by U(∑ c_{n}q^{n}) := ∑ c_{np}q^{n}. By [17, p. 458], \overline{f^{(−1)}(q)} is an eigenvector of T_{l} for every l ≠ p; moreover, \overline{f^{(−1)}(q)} ∈ ker U. Finally, for any open subscheme X′ ⊂ X_{1}(N)_{R} containing the ∞ section [∞] we have a natural injective q-expansion map O(X′ \ [∞])ˆ → R((q))ˆ , which we write as G ↦ G_{∞}. (See Section 4.4 for more details.)

Definition 3.16. An open subscheme of the form X′ \ [∞] with X′ as above will be called standard.

Let j(x) ∈ k be the j-invariant of x ∈ \overline{Y_{1}(N)}(k).

Theorem 3.17 (Explicit reciprocity functions for CL points). Assume, in Theorem 3.5, that X = S = X_{1}(N), Π = Id, and Φ is a modular parametrization attached to a newform f. Then one can choose X^{†} and Φ^{†} in Theorem 3.5 such that

(1) X^{†} is standard and X̄^{†} = \overline{Y_{1}(N)}^{ord} \ \{x | j(x) = 0, 1728\}.

(2) If A_{R} is not CL then Φ_{∞}^{†} = f^{(−1)}(q). In particular, \overline{Φ^{†}} = f̄^{(−1)}.

(3) If A_{R} is CL then Φ_{∞}^{†} = −uf_{[u]}^{(−1)}(q). In particular, (\overline{Φ^{†}})^{p} − ā_{p}\overline{Φ^{†}} = f̄^{(−1)}.

In both cases, (2) and (3), the function \overline{Φ^{†}} is integral over the integrally closed ring O(\overline{X_{1}(N)}^{ord}) and belongs to the fraction field of O(\overline{X_{1}(N)}^{ord}). So \overline{Φ^{†}} ∈ O(\overline{X_{1}(N)}^{ord}). Theorem 3.17 will be proved in Section 4.

Remark 3.18. If A_{R} is CL, then Theorem 3.17(3) implies that \overline{f_{[u]}^{(−1)}(q)} is the Fourier expansion of a rational function on \overline{X_{1}(N)}, hence of a quotient α/β where α, β ∈ M_{ν} are modular forms defined over k of some weight ν. Is there a direct argument for this?

Corollary 3.19. Let Φ: X_{1}(N) → A be a modular parametrization and let Γ ≤ A(R) be a subgroup with r := rank_{p}^{A(R)}(Γ) < ∞. Then the set Φ(CL) ∩ Γ is finite of cardinality at most

[(2g − 2 + ν) · \frac{p^{2} − p}{2} · p^{r} + 2λ] λ,

where g is the genus of X_{1}(N), ν is the number of cusps of X_{1}(N), and λ is the degree of X_{1}(N) → X_{1}(1).

Proof. By Theorem 3.17 we have d_{1} = 1, d_{2} = λ, and d_{3} ≤ 2λ in (3.9). So it will be enough to check that

(3.20)  deg(\overline{Φ^{†}}) ≤ (2g − 2 + ν) · \frac{p^{2} − p}{2}.

Taking degrees in parts (2) and (3) of Theorem 3.17 yields either deg(\overline{Φ^{†}}) = deg(f̄^{(−1)}) or p deg(\overline{Φ^{†}}) = deg(f̄^{(−1)}). In both cases, deg(\overline{Φ^{†}}) ≤ deg(f̄^{(−1)}). Now (3.20) follows from the fact that the numerator and denominator of the fraction in (3.15) are sections of the line bundle (Ω^{1}(cusps))^{\frac{p^{2}−p}{2}} , where Ω^{1} is the cotangent bundle on \overline{X_{1}(N)}. □

We next discuss a uniqueness property for the function Φ^{†} in Theorem 3.17. Let S = X_{1}(N), let X^{†} ⊂ S be a standard open subscheme over R such that

(3.21)  X̄^{†} ⊂ \overline{Y_{1}(N)}^{ord} \ \{x|j(x) = 0, 1728\}

and define (3.22)

P := {P ∈ CL | P̄ is not in the isogeny class of any of the k-points of Y_{1}(N) \ X^{†}}.

Clearly \overline{P} is infinite. Let M be the algebraic closure of Q in K and let ℘ be the place of M above which pR lies. We have P ⊂ X^{†}(O_{M,℘}). Let f = ∑ a_{n}q^{n} be a newform. Let ∑ P_{i}^{(l)} − a_{l}P be the Hecke divisor on S(\overline{Q}) associated to any P ∈ P and any prime l ≠ p (see Section 3.2). Then P_{i}^{(l)} ∈ CL ∩X^{†}(O_{M,℘}). For d ∈ (Z/NZ)^{×}, let ⟨d⟩ be the diamond operator acting on \overline{X_{1}(N)} and on O(\overline{X_{1}(N)}^{ord}). Consider the k-linear space

(3.23)  F := \{\overline{Θ} ∈ O(\overline{X_{1}(N)}^{ord}) | ⟨d⟩\overline{Θ} = \overline{Θ} for all d ∈ (Z/NZ)^{×} and U\overline{Θ}(q) = 0\} ,

where \overline{Θ}(q) ∈ k[[q]] is the Fourier expansion of \overline{Θ}. Note that f̄^{(−1)} ∈ F.

Theorem 3.24 (Uniqueness of reciprocity functions for CL points). Let S = X_{1}(N), and let Φ: S → A be a modular parametrization attached to a newform f = ∑ a_{n}q^{n} and let p be a sufficiently large good prime. Assume that X^{†} ⊂ S is an open subscheme over R as in (3.21). Let P be as in (3.22). Then the following conditions on \overline{Θ} ∈ F are equivalent.

1) For any P_{1}, …, P_{n} ∈ CL ∩X^{†}(R) and any integers m_{1}, …, m_{n} we have

∑_{i=1}^{n} m_{i}Φ(P_{i}) ∈ A(R)_{p-div} ⟹ ∑_{i=1}^{n} m_{i}\overline{Θ}(P̄_{i}) = 0 ∈ k.

2) For any P ∈ P and any prime l ≠ p we have

∑ \overline{Θ}(P̄_{i}^{(l)}) − a_{l}\overline{Θ}(P̄) = 0.

3) \overline{Θ} = λ̄ · \overline{f^{(−1)}} for some λ̄ ∈ k.

Proof. Condition 1 implies condition 2 by (A.5). That condition 2 implies condition 3 will be proved in Section 4: see Lemma 4.83. Finally condition 3 implies condition 1 by Theorem 3.17. □

3.7. Reciprocity functions and finiteness for isogeny classes. Fix a set Σ of rational primes.

Suppose that S = X_{1}(N). Let B be a Z[1/N]-algebra. Let Q be a B-point of Y_{1}(N), represented by (E_{Q}, α_{Q}). The Σ-isogeny class (respectively, the prime-to-Σ isogeny class) of Q in S(B) is the set C = C_{Q} ⊂ S(B) of all B-points of Y_{1}(N) represented by (E_{Q′}, α_{Q′}) such that there exists an isogeny E_{Q} → E_{Q′} of degree divisible only by primes in Σ (respectively, outside Σ). We do not require the isogeny to be compatible with α_{Q} and α_{Q′}.

The definition for S = X^{D}(U) is similar. Let B be a Z[1/m]-algebra. Let Q ∈ S(B) be represented by (E_{Q}, i_{Q}, α_{Q}). The Σ-isogeny class (respectively, the prime-to-Σ isogeny class) of Q in S(B) is the set C = C_{Q} ⊂ S(B) of all B-points of S represented by (E_{Q′}, i_{Q′}, α_{Q′}) such that there exists an isogeny E_{Q} → E_{Q′}, compatible with the O_{D}-action, and of degree divisible only by primes in Σ (respectively, outside Σ). Again the isogeny need not respect α_{Q} and α_{Q′}.

Let now S be either X_{1}(N) or X^{D}(U) and let C be a Σ-isogeny class where p ∉ Σ or a prime to Σ isogeny class where p ∈ Σ. Say that C is ordinary (respectively CL) if it contains an ordinary point (respectively a CL point); in this case all points in C are ordinary (respectively, CL).

Theorem 3.25 (Reciprocity functions mod p for isogeny classes). Assume that S ⟵^{Π} X ⟶^{Φ} A is a modular-elliptic or Shimura-elliptic correspondence, assume that p is a sufficiently large good prime, and assume C is an ordinary prime-to-p isogeny class in S(R). Then there exist an affine dense open subscheme X^{†} ⊂ X, a (not necessarily connected) finite étale cover π: X̄^{‡} → X̄^{†} of degree p, a regular function \overline{Φ^{‡}} ∈ O(X̄^{‡}) that is non-constant on each component of X̄^{‡}, and a map σ: Π^{−1}(C) ∩ X^{†}(R) → X̄^{‡}(k) such that π(σ(P)) = P̄ for all P, and for any P_{1}, …, P_{n} ∈ Π^{−1}(C) ∩ X^{†}(R) and any m_{1}, …, m_{n} ∈ Z we have

(3.26)  ∑_{i=1}^{n} m_{i}Φ(P_{i}) ∈ A(R)_{p-div} ⟺ ∑_{i=1}^{n} m_{i}\overline{Φ^{‡}}(σ(P_{i})) = 0 ∈ k.

Theorem 3.25 will be proved in Section 4.

Remark 3.27. 1) Again, as the proof will show, the maps \overline{Φ^{‡}} and σ will have a functorial nature. In Theorem 3.25 σ is simply a map of sets, but the proof will show that σ has actually an algebro-geometric flavor.

2) Theorem 3.25 is an analogue of the second equivalence in Theorem 3.5. Is there also an isogeny-class analogue of the first equivalence in Theorem 3.5?

3) The sum in the right half of (3.26) may be viewed as a function η^{‡} on \overline{X^{‡}}^{n} evaluated at (σ(P_{1}), …, σ(P_{n})). If the value is zero, then so is η^{†}(P̄_{1}, …, P̄_{n}), where η^{†} is the norm of η^{‡} in the degree-p^{n} extension O (\overline{X^{‡}}^{n}) of O (\overline{X^{†}}^{n}). Here η^{†} may be expressed as a polynomial in the m_{i} and the coefficients of the characteristic polynomial of multiplication-by-Φ^{‡} on the locally free O(\overline{X^{†}})-algebra O(\overline{X^{‡}}). Thus the left half of (3.26) implies a statement expressible in terms of evaluation of functions on \overline{X^{†}} instead of \overline{X^{‡}}. Theorem 3.32(4) will show that η^{†} is not always zero (consider the case n = 1, for example), so the statement is not always vacuous.

Theorem 3.25 trivially implies Corollary 3.28. In the notation of Theorem 3.25 we have

rank_{p}^{A(R)} (∑_{i=1}^{n} Z · Φ(P_{i})) = dim_{F_{p}} (∑_{i=1}^{n} F_{p} · \overline{Φ^{‡}}(σ(P_{i}))) .

Just as Corollary 3.8 implied Theorem 1.3, Corollary 3.28 applied to subsets {P_{1}, …, P_{n}} of Φ^{−1}(Γ) ∩ Π^{−1}(C) ∩ X^{†}(R) implies the first conclusion in

Corollary 3.29. Assume S ⟵^{Π} X ⟶^{Φ} A is a modular-elliptic or a Shimura-elliptic correspondence and assume p is a sufficiently big, good prime. Let C be an ordinary prime-to-p isogeny class in S(R). Then there exists a constant c such that for any subgroup Γ ≤ A(R) with r := rank_{p}^{A(R)}(Γ) < ∞ the set \overline{Φ(Π^{−1}(C)) ∩ Γ} ⊆ A(k) is finite of cardinality at most cp^{r}. In particular, the set Φ(Π^{−1}(C)) ∩ A(R)_{tors} is finite.

The first conclusion of Corollary 3.29 implies the last because the reduction map A(R)_{tors} → A(k) is injective for large p.

One can ask if the set Φ(Π^{−1}(C)) ∩ Γ is finite for every Γ with rank_{p}^{A(R)}(Γ) < ∞. Theorem 1.6 represents a partial result in this direction, with certain Σ-isogeny classes instead of prime-to-p isogeny classes. Corollary 3.29 will be used to prove Theorem 1.6 in Section 4.9.

3.8. Refinement of results on isogeny classes for modular parametrizations. Suppose that S ⟵^{Π} X ⟶^{Φ} A arises from a newform f = ∑ a_{n}q^{n}. Our goal in this subsection is to state Theorem 3.32, which describes the cover \overline{X^{‡}} and the function \overline{Φ^{‡}} explicitly in this case.

Let I_{1}(N) be the Igusa curve from pp. 460–461 of [17], except that we view I_{1}(N) as a smooth projective integral curve. It is a Galois cover of \overline{X_{1}(N)} ramified only over supersingular points, and the Galois group is naturally isomorphic to F_{p}^{×}. Let J := I_{1}(N)/⟨−1⟩ be the intermediate cover of degree (p − 1)/2 obtained by taking the quotient of I_{1}(N) by the involution corresponding to −1 ∈ F_{p}^{×}. We will describe \overline{X^{‡}} in terms of J. There is a point ∞ on each of these covers that is unramified over ∞ ∈ \overline{X_{1}(N)}. In particular, rational functions on I_{1}(N) and J have Fourier expansions in k((q)).

Let

(3.30)  f^{(0)}(q) := ∑_{(n,p)=1} a_{n}q^{n} ∈ Z_{p}[[q]].

(The series f^{(0)}(q) is called f|R_{0} in [38, p. 115].) Let

(3.31)  f_{[a_{p}]}^{(0)}(q) := (∑_{i=0}^{∞} a_{p}^{i}V^{i}) f^{(0)}(q) = ∑_{i=0}^{∞} ∑_{(n,p)=1} a_{p}^{i}a_{n}q^{np^{i}} ∈ Z_{p}[[q]].

Corollary 4.50 and Lemma 4.52 will show that for p ≫ 0, the series \overline{f_{[a_{p}]}^{(0)}(q)} is the Fourier expansion of some η ∈ k(J). For a constant λ̄ ∈ k to be specified later, define

\overline{Φ^{††}} := \begin{cases}λ̄η^{p^{2}} − ā_{p}η^{p}, & if A_{R} is not CL \\ η^{p}, & if A_{R} is CL. \end{cases}

Theorem 3.32 (Explicit reciprocity functions mod p for isogeny classes). Assume, in Theorem 3.25, that S ⟵^{Π} X ⟶^{Φ} A arises from a modular parametrization attached to a newform f on Γ_{0}(N). Then there exists λ̄ ∈ k^{×} such that X^{†}, \overline{X^{‡}}, and \overline{Φ^{‡}} can be chosen to satisfy:

1) The cover \overline{X^{‡}} of \overline{X^{†}} is a disjoint union \overline{X^{0}} ∐ \overline{X^{+}} ∐ \overline{X^{−}}, where \overline{X^{0}} ≃ \overline{X^{†}} is the trivial cover and \overline{X^{+}} and \overline{X^{−}} are each isomorphic to the inverse image of \overline{X^{†}} under J → \overline{X_{1}(N)}.

2) The restrictions of \overline{Φ^{‡}} to \overline{X^{0}}, \overline{X^{+}}, \overline{X^{−}} equal

\overline{Φ^{†}},  \overline{Φ^{†}} + λ_{+}\overline{Φ^{††}},  \overline{Φ^{†}} + λ_{−}\overline{Φ^{††}},

respectively, where λ_{±} ∈ k are such that λ_{+}^{(p−1)/2}, λ_{−}^{(p−1)/2} are the two square roots of λ̄.

Theorem 3.32 will be proved in Section 4.

Corollary 3.33. Let notation be as in Theorem 3.32. The characteristic polynomial of the endomorphism “multiplication by \overline{Φ^{‡}}” in the locally free O(\overline{X^{†}})-algebra O(\overline{X^{‡}}) is

x^{p} − λ̄h^{2}x + (λ̄h^{2}\overline{Φ^{†}} − (\overline{Φ^{†}})^{p}),

where h := (\overline{Φ^{††}})^{(p−1)/2} ∈ k(\overline{X_{1}(N)}).

Proof. The characteristic polynomial of \overline{Φ^{‡}} − \overline{Φ^{†}} equals

x (x^{(p−1)/2} − λ_{+}^{(p−1)/2}\overline{Φ^{††}}^{(p−1)/2}) (x^{(p−1)/2} − λ_{−}^{(p−1)/2}\overline{Φ^{††}}^{(p−1)/2}) = x^{p} − λ̄h^{2}x.

In this, replace x by x − \overline{Φ^{†}}. □

3.9. Strategy of proofs. The proof of our local results will be an application of the theory of δ-characters [3, 4] and δ-modular forms [5, 6]. These two types of objects are special cases of arithmetic differential equations in the sense of [7]. Section 4 reviews the facts from this theory that are necessary for the proof. As a sample of our strategy let us explain, very roughly, the idea of our proof of Theorem 3.5. Assume for simplicity that we are dealing with a modular-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A arising from a modular parametrization attached to a newform f. Following [3] consider the Fermat quotient operator δ: R → R defined by δx := (ϕ(x) − x^{p})/p, where ϕ: R → R is the lift of Frobenius. We view δ as an analogue of a derivation operator with respect to p. Recall from [3] that if Y is any smooth scheme over R then a function g: Y (R) → R is called a δ-function of order r if it is Zariski locally of the form P ↦ G(x, δx, …, δ^{r}x), where G is a restricted power series with R-coefficients and x ∈ R^{N} is a tuple of affine coordinates of P in some Ndimensional affine space. If A is our elliptic curve then by [3] there exists a δ-function of order 2, ψ: A(R) → R, which is also a group homomorphism; ψ is called in [3] a δ-character and may be viewed as an arithmetic analogue of the “Manin map” [25, 26]. Consider the composition f^{♯} = ψ ∘ Φ: X(R) → R. On the other hand, the theory of δ-modular forms [5] yields an open subset X^{†} of S and a δ-function of order 1, f^{♭}: X^{†}(R) → R, that vanishes at all CL-points: see Lemma 4.37 and (4.39). Then we prove that there exist δ-functions of order 2, denoted h_{0}, h_{1} : X^{†}(R) → R, such that the δ-function

Φ^{†} := f^{♯} − h_{0} · f^{♭} − h_{1} · δ ∘ f^{♭}

has order 0, or equivalently is a formal function in the usual sense of algebraic geometry. (Intuitively, in the system of “arithmetic differential equations” f^{♯} = f^{♭} = 0 one can eliminate all the “derivatives” of the unknowns.) It follows that f^{♯} and Φ^{†} have the same value at each CL-point P_{i}. So

∑ m_{i}Φ^{†}(P_{i}) = ∑ m_{i}f^{♯}(P_{i}) = ψ(∑ m_{i}Φ(P_{i})).

By the arithmetic analogue in [3, 4] of Manin’s Theorem of the Kernel [25, 26], ψ(∑ m_{i}Φ(P_{i})) vanishes if and only if ∑ m_{i}Φ(P_{i}) is torsion. (Actually, for our application to Theorem 1.3 we need only the “if” part, which does not require the analogue of the Theorem of the Kernel.) On the other hand we will check that \overline{Φ^{†}} ∉ k by looking at Fourier q-expansions, and this will complete the proof of the first equivalence in Theorem 3.5 in the special case we considered.

In particular, our proof of the (effective) finiteness of Φ(CL) ∩ Γ in the case Γ = A(R)_{tors} can be intuitively described as follows. The points of CL are solutions of the “arithmetic differential equation” f^{♭} = 0 whereas the points of Φ^{−1}(Γ) are solutions of the “arithmetic differential equation” f^{♯} = 0. Hence the points of CL ∩Φ^{−1}(Γ) are solutions of the system of “arithmetic differential equations” f^{♭} = f^{♯} = 0. By what was said above one can eliminate, in this system, the “derivatives” of the unknowns and hence one is left with a (non-differential) algebraic equation mod p, whose “degree” can be estimated. There are only finitely many solutions to this algebraic equation and their number is effectively bounded by the “degree”.

4. Proofs

Fix a prime p ≥ 5. Recall that R = Ẑ_{p}^{ur}, k = R/pR, K := R[1/p], and ϕ: R → R is the Frobenius automorphism.

4.1. Review of CL and CM points. This section reviews facts we need about CL abelian schemes and their relation with CM points; see [22, 15, 29]. Expert readers should skip this discussion.

Definition 4.1. An abelian scheme E/R is CL (a canonical lift) if its reduction Ē := E ⊗ k is ordinary and there exists an R-homomorphism E → E^{ϕ} := E ⊗_{R,ϕ} R whose reduction mod p is the relative Frobenius k-homomorphism Ē → Ē^{Fr} := Ē ⊗_{k,Fr} k.

Theorem 4.2. The following are equivalent for an elliptic curve E over R:

(1) E is CL.

(2) E has ordinary reduction and Serre-Tate parameter q(E) = 1 (with respect to some, and hence any, basis of the physical Tate module).

(3) There exists a morphism of Z-schemes E → E whose reduction mod p is the absolute Frobenius F_{p}-morphism Ē → Ē. (In [7] this situation was referred to by saying that E has a lift of Frobenius.)

Proof. The equivalence between 2 and 1 is essentially the definition of the CL property in [22]. The implication 1 ⟹ 3 is trivial. For 3 ⟹ 1, note first that Ē must be ordinary: this follows, for instance, from Proposition 7.15 and Corollaries 8.86 and 8.89 in [7]. Finally, the Z-morphism E → E induces an R-morphism E → E^{ϕ}; the Néron model property shows that the latter is a composition of a homomorphism u with translation by an R-point reducing to the identity mod p. But then u mod p is the relative Frobenius. □

Theorem 4.3 (Existence and uniqueness of CL abelian schemes).

(1) Fix a prime p and an ordinary abelian variety Ē over k. Then there exists a unique CL abelian scheme E over R with E ⊗ k ≃ Ē (unique up to isomorphism).

(2) If E and E′ are CL abelian schemes over R, then the natural map Hom_{R}(E, E′) → Hom_{k}(Ē, Ē′) is an isomorphism.

(3) If two elliptic curves over R are related by an isogeny of degree prime to p and one of them is CL, then so is the other.

Proof. This is due to Serre and Tate: see [22, 15]. □

The conductor of an order in a quadratic number field is the index of the order in the maximal order.

Theorem 4.4 (Relation between CL and CM).

(1) (a) If E is a CL elliptic curve over R, then E has CM (part of this claim is that E is definable over M = K ∩ \overline{Q}). Thus we have the relation CL ⊆ CM between subsets of Y_{1}(N)(\overline{Q}).

(b) Conversely, if Q = (E, α) ∈ Y_{1}(N)(\overline{Q}) is in CM, and p is split in End E ⊗ Q

and does not divide the conductor of End E, then Q ∈ CL.

(2) (a) If (E, i) is a CL fake elliptic curve over R, then (E, i) is CM. Thus we have the

relation CL ⊆ CM between subsets of X^{D}(U)(M).

(b) Conversely, for any CM-point Q ∈ X^{D}(U)(\overline{Q}), we know that the associated abelian surface E is the square of an elliptic curve with CM by an order in some K; if p splits in K and p does not divide the conductor of the order, then Q ∈ CL.

Proof.

(1) (a) If E/R is a CL elliptic curve, then End_{R}(E) ≃ End_{k}(Ē) ≠ Z.

(b) This follows from the theorem in the middle of p. 293 in [36].

(2) (a) Let E := End_{R}(E) ⊗ Q ≃ End_{k}(Ē) ⊗ Q. Since Ē is ordinary, the center of E contains an imaginary quadratic field K: see [7, p. 247], say. In particular, E ≄ D, so (E, i) is CM.

(b) Apply Theorem 4.4(1)(b) to the elliptic curve.

□

4.2. δ-functions. See [3, 7]. Let δ: R → R be the Fermat quotient map δx := (ϕ(x)−x^{p})/p. Then

(4.5)  \begin{aligned}δ(x + y) & = & δx + δy + C_{p}(x, y) \\ δ(xy) & = & x^{p} · δy + y^{p} · δx + p · δx · δy, \end{aligned}

where C_{p}(X, Y) := \frac{X^{p}+Y^{p}−(X+Y)^{p}}{p} ∈ Z[X, Y]. Following [3] we think of δ as a “derivation with respect to p”. If P ∈ A^{N}(R) = R^{N}, then δP is defined by applying δ to each coordinate.

Let X be a smooth R-scheme and let f: X(R) → R be a map of sets. Following [7, p. 41], we say that f is a δ-function of order r if for any point in X(R) there is a Zariski open neighborhood U ⊂ X, a closed immersion u: U ↪ A_{R}^{N}, and a restricted power series F with R-coefficients in (r + 1)N variables such that

f(P) = F (u(P), δ(u(P)), …, δ^{r}(u(P))) for all P ∈ U(R).

(Restricted means that the coefficients converge p-adically to 0.) Let O^{r}(X) be the ring of δ-functions of order r on X.

We have natural maps δ: O^{r}(X) → O^{r+1}(X), f ↦ δf := δ ∘ f, and natural ring homomorphisms ϕ: O^{r}(X) → O^{r+1}(X), f ↦ ϕ(f) = f^{ϕ} := ϕ ∘ f. The maps δ above still satisfy the identities in (4.5). Let X be affine, and let x be a system of étale coordinates on X, that is to say there exists an étale map X → A^{d} such that x is the d-tuple of elements in O(X) obtained by pulling back the coordinates on A^{d}. Let x′, x′′, …, x^{(r)} be d-tuples of variables and let ˆ denotes p-adic completion, as usual. Then the natural map

(4.6)  O(X)ˆ [x′, x′′, …, x^{(r)}]ˆ → O^{r}(X)

sending x′ ↦ δx, x′′ ↦ δ^{2}x,…,x^{(r)} ↦ δ^{r}x is an isomorphism: see Propositions 3.13 and 3.19 in [7].

4.3. δ-characters. We recall facts from [3, 7]. If G is a smooth group scheme over R, then by a δ-character of order r we understand a δ-function ψ: G(R) → R of order r which is also a group homomorphism into the additive group of R. Following [3], we view δ-characters of abelian schemes as arithmetic analogues of the Manin maps [25, 26]. Let X^{r}(G) be the R-module of δ-characters of order r on G. By [3, pp. 325-326], the following hold for an elliptic curve E/R:

(1) If E is CL, then X^{1}(E) is free of rank 1.

(2) If E is not CL, then X^{2}(E) is free of rank 1. We will need to review (and complement) some results in [3, 4] that can be viewed as an arithmetic analogue of Manin’s Theorem of the Kernel [26, 10]. For any abelian group G, we set p^{∞}G := ∩_{n=1}^{∞}p^{n}G and we let p^{∞}G : p^{∞} be the group of all x ∈ G for which there exists an integer n ≥ 1 with p^{n}x ∈ p^{∞}G. Also recall that we set G_{p-div} = G_{tors} + pG.

Lemma 4.7. Let E be an elliptic curve over Z_{p}. Let r be 1 or 2 according as E is CL or not. Let ψ: E(R) → R be a generator of X^{r}(G). Then

(1) ψ is surjective and defined over Z_{p}.

(2) ker ψ = p^{∞}E(R) : p^{∞}.

(3) ker ψ + pE(R) = E(R)_{tors} + pE(R) =: E(R)_{p-div}.

(4) ψ^{−1}(pR) = E(R)_{tors} + pE(R) =: E(R)_{p-div}.

(5) (ker ψ) ∩ E(Z_{p}^{ur}) = E(Z_{p}^{ur})_{tors}.

Proof.

(1) Surjectivity follows from [4, Theorem 1.10]. That ψ is defined over Z_{p} follows from its construction in [3].

(2) If E has ordinary reduction, then [3, Theorem B’, p. 312] shows that (ker ψ)/p^{∞}E(R) is a finite cyclic p-group; this implies the non-trivial inclusion “⊂”. If E has supersingular reduction, we are done by [4, Corollary 1.12].

(3) The non-trivial inclusion is “⊂”. If P ∈ ker ψ, by (2) there exists n such that p^{n}P = p^{n+1}Q for some Q ∈ E(R). So P − pQ ∈ E(R)_{tors} and we are done.

(4) This follows from (3) and (1).

(5) If E has ordinary reduction, then by Theorem 1.2 and Remark 1.3 on p. 209 of [4], we have p^{∞}E(R) ∩ E(Z_{p}^{ur}) ⊂ E(Z_{p}^{ur})_{tors}; combining this with (2) yields the nontrivial inclusion (ker ψ) ∩ E(Z_{p}^{ur}) ⊂ E(Z_{p}^{ur})_{tors} of (5). Now assume that E has supersingular reduction. If a_{p} is the trace of Frobenius on E_{F_{p}} then the map ϕ^{2} − a_{p}ϕ + p: R → R is injective. By [4, Theorem 1.10, p. 212], the restriction of ψ to the kernel of the reduction map red: E(R) → E(k) is injective. In other words, ker(ψ) ∩ ker(red) = {0}. Equivalently, red restricts to an injection ker ψ → E(k). Since E(k) is torsion, so is ker ψ.

□

We now describe an explicit generator ψ of X^{r}(A_{R}), where A is an elliptic curve over Z_{p}, and r is 1 or 2 according as A_{R} is CL or not. Fix a 1-form ω generating the Z_{p}-module H^{0}(A, Ω^{1}). This uniquely specifies a Weierstrass model y^{2} = x^{3} + ax + b for A over Z_{p} such that ω = dx/y. Let T := −x/y. So T is an étale coordinate at the origin 0 of A, vanishing at 0. Let L(T) ∈ Q_{p}[[T]] be the logarithm of the formal group of A associated to T, so dL(T) = ω ∈ Z_{p}[[T]] dT and L(0) = 0. If A is CL, let up be the unique root in pZ_{p} of the polynomial x^{2} − a_{p}x + p. By [7, Theorem 7.22] and [4, Theorem 1.10], we may take

(4.8)  ψ := \begin{cases}\frac{1}{p}(ϕ^{2} − a_{p}ϕ + p)L(T) ∈ R[[T]][T′, T′′]ˆ , & if A is not CL; \\ \frac{1}{p}(ϕ − up)L(T) ∈ R[[T]][T′]ˆ , & if A is CL. \end{cases}

4.4. δ-Fourier expansions. See [5]. We start by reviewing background on classical Fourier expansions as in [13, p. 112]. (The discussion there involves the modular curve parametrizing elliptic curves with an embedding of μ_{N} rather than Z/NZ as here. But, the two modular curves are isomorphic over Z[1/N, ζ_{N}]: see [13, p. 113].) The cusp ∞ on S := X_{1}(N) arises from a Z[1/N, ζ_{N}]-valued point; so if p ≫ 0 (specifically, p ∤ N), then it gives rise to an Rpoint, which may be viewed as a closed immersion s_{∞}: Spec R → S_{R}. Let [∞] = s_{∞}(Spec R). Let S̃_{R} be the completion of S_{R} along [∞]. The Tate generalized elliptic curve Tate(q)/R[[q]] equipped with the standard immersion α_{can} of μ_{N,R} ≃ (Z/NZ)_{R} is a point in S(R[[q]]) that reduces mod q to s_{∞}. For p ≫ 0 there is an induced isomorphism Spf R[[q]] ≃ S̃_{R}. Therefore, for any open subset U ⊂ S_{R} containing [∞] we have an induced Fourier qexpansion homomorphism

O(U \ [∞]) → R((q)) := R[[q]][1/q].

More generally, suppose that we are given a modular-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A. Let M be the ramification index of Π at x_{∞}. As before, we assume p ≫ 0. Then we have Spf R[[𝔮]] ≃ X̃_{R}, where 𝔮 := q^{1/M} and X̃_{R} is the completion of X_{R} along the closure [x_{∞}] of x_{∞}. Moreover, for any open set U ⊂ X_{R} containing [x_{∞}] we have a Fourier q-expansion homomorphism

O(U \ [x_{∞}]) → R((𝔮)).

Next we move to the “δ-theory”. Let q′, q′′, …, q^{(r)}, … be new indeterminates. Define

S_{∞}^{r} := R((q))ˆ [q′, q′′, …, q^{(r)}]ˆ .

For each r, extend ϕ: R → R to a ring homomorphism ϕ: S_{∞}^{r} → S_{∞}^{r+1} denoted F ↦ F^{ϕ} by requiring

q^{ϕ} := q^{p} + pq′, (q′)^{ϕ} := (q′)^{p} + pq′′, …,

and define δ: S_{∞}^{r} → S_{∞}^{r+1} by

(4.9)  δF := \frac{F^{ϕ} − F^{p}}{p} .

By the universality property of the sequence {O^{r}(U \ [∞])}_{r≥0} (see [7, Proposition 3.3]), there exists a unique sequence of ring homomorphisms

(4.10)  O^{r}(U \ [∞]) → S_{∞}^{r},

called δ-Fourier expansion maps and denoted g ↦ g_{∞}, such that (δg)_{∞} = δ(g_{∞}) for all g. More generally, given a modular-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A, define rings

S_{x_{∞}}^{r} := R((𝔮))ˆ [𝔮′, …, 𝔮^{(r)}]ˆ

where 𝔮′, …, 𝔮^{(r)} are new variables. Again there are natural maps ϕ, δ: S_{x_{∞}}^{r} → S_{x_{∞}}^{r+1} defined exactly as above and there are δ-Fourier expansion maps

O^{r}(U \ [x_{∞}]) → S_{x_{∞}}^{r}

commuting with δ, and denoted g ↦ g_{x_{∞}}. There are natural maps S_{∞}^{r} → S_{x_{∞}}^{r}. Since Spec R[𝔮, 𝔮^{−1}] → Spec R[q, q^{−1}] is étale, (4.6) implies

S_{x_{∞}}^{r} ≃ R((𝔮))ˆ [q′, …, q^{(r)}]ˆ .

4.5. δ-Serre-Tate expansions. See [6, 7]. Assume that we are given a Shimura-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A, and that p ≫ 0. By the proof of Lemma 2.6 in [6], there exist infinitely many k-points ȳ_{0} ∈ S(k) whose associated triple (Ȳ , ī, ᾱ) is such that

(1) Ȳ is ordinary, and

(2) if θ̄ is the unique principal polarization compatible with ī, then (Ȳ , θ̄) is isomorphic

to the polarized Jacobian of a genus-2 curve. So we may choose a point ȳ_{0} ∈ S(k) as above such that moreover, there exists x̄_{0} ∈ X̄(k) with Π(x̄_{0}) = ȳ_{0} such that both Π and Φ are étale at x̄_{0}: here we use p ≫ 0 to know that Π ⊗ k and Φ ⊗ k are separable.

Let Y be the canonical lift of Ȳ. Since End(Y) ≃ End(Ȳ), the embedding ī: O_{D} → End(Ȳ) induces an embedding i: O_{D} → End(Y). Also the level U structure ᾱ lifts to a level U structure on (Y, i). Let y_{0} := (Y, i, α) ∈ S(R). Since Π is étale at x̄_{0}, there exists x_{0} ∈ X(R) such that x_{0} mod p = x̄_{0} and Π(x_{0}) = y_{0}.

Let Ȳ^{∨} be the dual of Ȳ. By [6, Lemma 2.5], there exist Z_{p}-bases of the Tate modules T_{p}(Ȳ) and T_{p}(Ȳ^{∨}), corresponding to each other under θ̄, such that any fake elliptic curve over R lifting (Ȳ , ī) has a diagonal Serre-Tate matrix diag(q, q^{disc(D)}) with respect to these bases. Fix such bases. They define an isomorphism between the completion of S_{R} along the section y_{0} and Spf R[[t]]. The Serre-Tate parameter q corresponds to the value of 1 + t. Since Π is étale at x̄_{0} we have an induced isomorphism between the completion of X along the section x_{0} and Spf R[[t]]. As in Section 4.4 define rings

S_{x_{0}}^{r} ≃ R[[t]][t′, …, t^{(r)}]ˆ

and maps ϕ, δ: S_{x_{0}}^{r} → S_{x_{0}}^{r+1}; then for any affine open set U ⊂ X containing the image of the section x_{0} we have natural δ-Serre-Tate expansion maps

(4.11)  O^{r}(U) → S_{x_{0}}^{r},

denoted g ↦ g_{x_{0}}, that commute with ϕ and δ.

4.6. Pullbacks by Φ of δ-characters. Assume that we are given a modular-elliptic or a Shimura-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A. Recall that A is defined over a number field F_{0}. We suppose that p ≫ 0 and p splits completely in F_{0}. Then A_{R} comes from an elliptic curve over Z_{p}. Define a_{p} and (if A_{R} is CL) u as in Section 4.3. Let ψ be as in (4.8). The composition

(4.12)  f^{♯}: X(R) →^{Φ} A(R) →^{ψ} R.

is in O^{r}(X_{R}). In what follows we compute the δ-Fourier expansion f_{x_{∞}}^{♯} ∈ S_{x_{∞}}^{r} (in the modular-elliptic case) or the δ-Serre-Tate expansion f_{x_{0}}^{♯} ∈ S_{x_{0}}^{r} (in the Shimura-elliptic case).

4.6.1. Modular-elliptic case. Suppose that S = X_{1}(N). We have Φ^{∗}: R[[T]] → R[[𝔮]]. Define b_{n} ∈ F_{0} ∩ R by

(∑_{n≥1} b_{n}𝔮^{n−1}) d𝔮 := d(Φ^{∗}(L(T))) = Φ^{∗}(dL(T)) = Φ^{∗}ω.

so

(4.13)  ∑_{n≥1} \frac{b_{n}}{n} 𝔮^{n} = Φ^{∗}(L(T)).

Applying Φ^{∗} to (4.8) and substituting (4.13) yields

(4.14)  f_{x_{∞}}^{♯} = Φ^{∗}ψ = \begin{cases}\frac{1}{p} ∑_{n≥1} (\frac{b_{n}^{ϕ^{2}}}{n} 𝔮^{nϕ^{2}} − a_{p} \frac{b_{n}^{ϕ}}{n} 𝔮^{nϕ} + p\frac{b_{n}}{n} 𝔮^{n}) , & if A is not CL; \\ \frac{1}{p} ∑_{n≥1} (\frac{b_{n}^{ϕ}}{n} 𝔮^{nϕ} − up\frac{b_{n}}{n} 𝔮^{n}) , & if A is CL. \end{cases}

In both cases, f_{x_{∞}}^{♯} ∈ R[[𝔮]][𝔮′, 𝔮′′]ˆ . Applying the substitution homomorphism

R[[𝔮]][𝔮′, 𝔮′′]ˆ → R[[𝔮]]
G ↦ G_{♮} := G(𝔮, 0, 0) = G|_{𝔮′=𝔮′′=0},

we obtain

(4.15)  (f_{x_{∞}}^{♯})_{♮} = \begin{cases}\frac{1}{p} ∑_{n≥1} (^{\frac{b_{n/p^{2}}^{ϕ^{2}}}{n/p^{2}}} − a_{p}^{\frac{b_{n/p}^{ϕ}}{n/p}} + p\frac{b_{n}}{n} ) q_{M}^{n}, & if A is not CL; \\ \frac{1}{p} ∑_{n≥1} (^{\frac{b_{n/p}^{ϕ}}{n/p}} − up\frac{b_{n}}{n} ) 𝔮^{n}, & if A is CL, \end{cases}

where b_{γ} := 0 if γ ∈ Q \ Z. (In particular, the right hand side of (4.15) has coefficients in R, which is not a priori obvious.)

Let us consider the special case when S ⟵^{Π} X ⟶^{Φ} A arises from a modular parametrization associated to the newform f = ∑ a_{n}q^{n}, so S = X = X_{1}(N), Π = Id, x_{∞} = ∞, M = 1, and 𝔮 = q. We may take ω so that Φ^{∗}ω = ∑ a_{n}q^{n−1}dq; then b_{n} = a_{n} for all n. Since f is a newform, the a_{n} satisfy the usual relations [41, Theorem 3.43] (we use p ≫ 0 to know that p ∤ N):

(4.16)  a_{p^{i}m} = a_{p^{i}}a_{m} for (p, m) = 1,
(4.17)  a_{p^{i−1}}a_{p} = a_{p^{i}} + pa_{p^{i−2}} for i ≥ 2.

Lemma 4.18. Assume that S ⟵^{Π} X ⟶^{Φ} A arises from a modular parametrization attached to f.

(1) With notation as in (3.12) and (3.13), the following holds in Z_{p}[[q]]:

(4.19)  (f_{∞}^{♯})_{♮} = \begin{cases}f^{(−1)}(q), & if A is not CL; \\ −uf_{[u]}^{(−1)}(q), & if A is CL. \end{cases}

(2) With notation as in (3.30) and (3.31), the following holds in k[[q]][q′, q′′]:

(4.20) \overline{f_{∞}^{♯}} = \begin{cases}\overline{f^{(−1)}(q)} + (\frac{q′}{q^{p}} )^{p} (\overline{f_{[a_{p}]}^{(0)}(q)})^{p^{2}} − ā_{p} (\frac{q′}{q^{p}} ) (\overline{f_{[a_{p}]}^{(0)}(q)})^{p} , & if A is not CL; \\ −ū\overline{f_{[u]}^{(−1)}(q)} + (\frac{q′}{q^{p}} ) (\overline{f_{[a_{p}]}^{(0)}(q)})^{p} , & if A is CL. \end{cases}

Proof. We shall prove (4.20) in the case where A_{R} is not CL. The other three statements are proved similarly (and are actually easier).

To simplify notation, let □ stand for any element of Z_{p}[[q]][q^{−1}, q′, q′′]ˆ . For any γ, β ∈ Z_{p}[[q]][q^{−1}, q′, q′′]ˆ , any ℓ ∈ Z_{≥2}, and any m ∈ Z_{≥1} we have

(4.21)  (1 + pγ + p^{2}β)^{mp^{ℓ−2}} = 1 + mp^{ℓ−1}γ + p^{ℓ}□.

(Writing (1 + pγ + p^{2}β)^{m} as 1 + pγ′ lets us reduce to the case β = 0 and m = 1, which is proved by induction on ℓ.)

By (4.14) we get

f_{∞}^{♯} = \frac{1}{p} [∑ \frac{a_{n}}{n} (q^{p^{2}} + p(q′)^{p} + p^{2}□)^{n} − a_{p} ∑ \frac{a_{n}}{n} (q^{p} + pq′)^{n} + p ∑ \frac{a_{n}}{n} q^{n}]

= \frac{1}{p} [∑ \frac{a_{n}}{n} (1 + p(\frac{q′}{q^{p}})^{p} + p^{2}□)^{n} q^{p^{2}n} − a_{p} ∑ \frac{a_{n}}{n} (1 + p\frac{q′}{q^{p}} )^{n} q^{pn} + p ∑ \frac{a_{n}}{n} q^{n}]

= ∑ [\frac{a_{n/p^{2}}}{n/p} (1 + p (\frac{q′}{q^{p}} )^{p} + p^{2}□)^{n/p^{2}} − a_{p} \frac{a_{n/p}}{n} (1 + p\frac{q′}{q^{p}} )^{n/p} + \frac{a_{n}}{n} ] q^{n}

=: ∑ γ_{n}q^{n},

where a_{r} = 0 for r ∈ Q \ Z.

If (n, p) = 1, then γ_{n} = a_{n}/n.

If n = pm with (m, p) = 1, then (4.16) and (4.21) yield

γ_{n} = \frac{a_{p}a_{m}}{pm} − a_{p} \frac{a_{m}}{pm} (1 + pm\frac{q′}{q^{p}} + p^{2}□) ≡ −a_{p}a_{m} \frac{q′}{q^{p}} (mod p).

If n = p^{ℓ}m with ℓ ≥ 2 and (m, p) = 1, then (4.16), (4.17), and (4.21) yield

γ_{n} = \frac{a_{p^{ℓ−2}}a_{m}}{p^{ℓ−1}m} (1 + mp^{ℓ−1} (\frac{q′}{q^{p}} )^{p} + p^{ℓ}□) − \frac{a_{p}a_{p^{ℓ−1}}a_{m}}{p^{ℓ}m} (1 + mp^{ℓ} \frac{q′}{q^{p}} + p^{ℓ+1}□) + \frac{a_{p^{ℓ}}a_{m}}{p^{ℓ}m}

≡ a_{p}^{ℓ−2}a_{m} (\frac{q′}{q^{p}} )^{p} − a_{p}^{ℓ}a_{m} \frac{q′}{q^{p}} (mod p).

Therefore

f_{∞}^{♯} ≡ ∑_{(m,p)=1} \frac{a_{m}}{m} q^{m}−a_{p} \frac{q′}{q^{p}} ∑_{(m,p)=1} a_{m}q^{mp}+∑_{ℓ≥2} ∑_{(m,p)=1} a_{m} (a_{p}^{ℓ−2} (\frac{q′}{q^{p}} )^{p} − a_{p}^{ℓ} \frac{q′}{q^{p}} ) q^{mp^{ℓ}} (mod p),

and the first case of (4.20) follows via a trivial algebraic manipulation. □

Remark 4.22. The right hand side of (4.20) belongs to the subring k[[q]][q′] of k[[q]][q′, q′′]. In the case where S ⟵^{Π} X ⟶^{Φ} A does not necessarily arise from a modular parametrization, an argument similar to the one in the proof of Lemma 4.18 still yields

(4.23)  \overline{f_{x_{∞}}^{♯}} ∈ k[[𝔮]][𝔮′].

4.6.2. Shimura-elliptic case. Suppose that S = X^{D}(U). Recall that we fixed x_{0} ∈ X(R) and a corresponding δ-Serre-Tate expansion map O^{2}(X_{R}) → S_{x_{0}}^{2} = R[[t]][t′, t′′]ˆ , denoted G ↦ G_{x_{0}}. Let z_{0} = Φ(x_{0}) ∈ A(R). Let λ: A_{R} → A_{R} be the translation by −z_{0}. Recall the étale coordinate T on A_{R} at 0; use T_{z_{0}} := λ^{∗}T as étale coordinate at z_{0}. Now we have R[[T]] →^{λ^{∗}} R[[T_{z_{0}}]] →^{Φ^{∗}} R[[t]]. Define b_{n} ∈ F_{0} ∩ R by

(∑_{n≥1} b_{n}t^{n−1}) dt := d(Φ^{∗}λ^{∗}(L(T))) = Φ^{∗}λ^{∗}d(L(T)) = Φ^{∗}λ^{∗}ω,

so

(4.24)  ∑_{n≥1} \frac{b_{n}}{n} t^{n} = Φ^{∗}λ^{∗}(L(T)).

Since Φ is étale at x_{0}, we have b_{1} ≠ 0; scaling ω, we may assume that b_{1} = 1. Since ψ is a group homomorphism, we have ψ − ψ(z_{0}) = λ^{∗}ψ. Add the constant ψ(z_{0}) to both sides, and apply Φ^{∗} to obtain

f_{x_{0}}^{♯} = Φ^{∗}ψ = ψ(z_{0}) + Φ^{∗}λ^{∗}ψ.

Evaluate Φ^{∗}λ^{∗}ψ by applying Φ^{∗}λ^{∗} to (4.8) and substituting (4.24) into the right hand side: the final result is

(4.25)  f_{x_{0}}^{♯} = \begin{cases}ψ(z_{0}) + \frac{1}{p} ∑_{n≥1} (\frac{b_{n}^{ϕ^{2}}}{n} t^{nϕ^{2}} − a_{p} \frac{b_{n}^{ϕ}}{n} t^{nϕ} + p\frac{b_{n}}{n} t^{n}) , & if A is not CL; \\ ψ(z_{0}) + \frac{1}{p} ∑_{n≥1} (\frac{b_{n}^{ϕ}}{n} t^{nϕ} − up\frac{b_{n}}{n} t^{n}) , & if A is CL. \end{cases}

An argument similar to the one in the proof of Lemma 4.18 shows that

(4.26)  \overline{f_{x_{0}}^{♯}} ∈ k[[t]][t′].

4.7. δ-modular forms: modular-elliptic case. We recall some concepts from [5, 7, 1]. The ring of δ-modular functions [5] is

M^{r} := R[a_{4}^{(≤r)}, a_{6}^{(≤r)}, Δ^{−1}]ˆ ,

where a_{4}^{(≤r)} is a tuple of variables (a_{4}, a′_{4}, a′′_{4}, …, a_{4}^{(r)}) and a_{6}^{(≤r)} is similar, and Δ := −2^{6}a_{4}^{3} − 2^{4}3^{3}a_{6}^{2}. If g ∈ M^{0} \ pM^{0}, define

M_{{g}}^{r} := M^{r}[g^{−1}]ˆ = R[a_{4}^{(≤r)}, a_{6}^{(≤r)}, Δ^{−1}, g^{−1}]ˆ .

An element of M^{r} or M_{{g}}^{r} is defined over Z_{p} if it belongs to the analogously defined ring with Z_{p} in place of R. Define δ: M^{r} → M^{r+1} and δ: M_{{g}}^{r} → M_{{g}}^{r+1} as δ: S_{∞}^{r} → S_{∞}^{r+1} was defined in Section 4.4. Let j : −2^{12}3^{3}a_{4}^{3}/Δ, let i := 2^{6}3^{3} − j, and let t := a_{6}/a_{4}. (This t is unrelated to the t used in δ-Serre-Tate expansions.) By [5, Proposition 3.10], we have

M_{{a_{4}a_{6}}}^{r} = R[j^{(≤n)}, j^{−1}, i^{−1}, t^{(≤r)}, t^{−1}]ˆ .

If w = ∑ n_{i}ϕ^{i} ∈ Z[ϕ], define deg w = ∑ n_{i}. If moreover λ ∈ R, define λ^{w} := ∏(λ^{ϕ^{i}})^{n_{i}}. For w ∈ Z[ϕ], say that f in M^{r} or M_{{g}}^{r} is of weight w if

(4.27)  f(λ^{4}a_{4}, λ^{6}a_{6}, δ(λ^{4}a_{4}), δ(λ^{6}a_{6}), …) = λ^{w}f(a_{4}, a_{6}, a′_{4}, a′_{6}, …),

for all λ ∈ R. Let M^{r}(w) be the set of f ∈ M^{r} of weight w, and define M_{{g}}^{r}(w) similarly. In [5], elements of M_{{g}}^{r}(w) were called δ-modular forms of weight w (holomorphic outside g = 0).

If f ∈ M_{{g}}^{r}(w) and E is an elliptic curve given by y^{2} = x^{3} + Ax + B with A, B ∈ R and g(A, B) ∈ R^{×}, then define f(A, B) ∈ R by making the substitutions a_{4} ↦ A, a_{6} ↦ B, a′_{4} ↦ δA, a′_{6} ↦ δB, a′′_{4} ↦ δ^{2}A, and so on. Recall from [5] that f is called isogeny covariant if for any isogeny u of degree prime to p from an elliptic curve y^{2} = x^{3} + A_{1}x + B_{1} with g(A, B) ∈ R^{×} to an elliptic curve y^{2} = x^{3} + A_{2}x + B_{2} with g(A_{2}, B_{2}) ∈ R^{×} that pulls back dx/y to dx/y we have

f(A_{1}, B_{1}) = deg(u)^{− deg(w)/2}f(A_{2}, B_{2}).

By [5, Corollary 3.11], M_{{a_{4}a_{6}}}^{r}(0) = R[j^{(≤r)}, j^{−1}, i^{−1}]ˆ . More generally, if m ∈ 2Z and g ∈ M^{0}(m), define g̃ := gt^{−m/2}; then

(4.28)  M_{{a_{4}a_{6}g}}^{r}(0) = R[j^{(≤r)}, j^{−1}, i^{−1}, g̃^{−1}]ˆ .

Also define the open subscheme Y (1)^{g} := Spec R[j, j^{−1}, i^{−1}, g̃] of the modular curve Y (1)_{R} := Spec R[j]. If we define

(4.29)  b := a_{6}^{2}/a_{4}^{3} = −2^{2}3^{−3} + 2^{8}j^{−1}.

then R[j, j^{−1}, i^{−1}] = R[b, b^{−1}, (4+27b)^{−1}], so b is an étale coordinate on Y (1)^{g}, and Y_{1}(N)_{R} → Y (1)_{R} is étale over Y (1)^{g}. Suppose that in addition we are given a modular-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A. Then we may (and do) choose g so that the composition v: X_{R} →^{Π} X_{1}(N)_{R} → X(1)_{R} is étale above Y (1)^{g}. Set

(4.30)  X^{†} := v^{−1}(Y (1)^{g}).

The pull-back of b to X^{†}, which we will still call b, is an étale coordinate on X^{†}. By (4.6), we have natural isomorphisms

(4.31)  O(X^{†})ˆ [b′, …, b^{(r)}]ˆ ≃ O^{r}(X^{†}),

where b′, …, b^{(r)} are new indeterminates. We view (4.31) as an identification. Similarly, since j is an étale coordinate on Y (1), (4.6) and (4.28) yield

(4.32)  M_{{a_{4}a_{6}g}}^{r}(0) ≃ O^{r}(Y (1)^{g}) ⊂ O^{r}(X^{†}).

Since X^{†} is standard in the sense of Definition 3.16, we have the δ-Fourier expansion map

(4.33)  O^{r}(X^{†}) → S_{x_{∞}}^{r}.

Composing (4.32) and (4.33) yields δ-Fourier expansion maps

(4.34)  M_{{a_{4}a_{6}g}}^{r}(0) → S_{x_{∞}}^{r}.

Let E_{4}(q) and E_{6}(q) be the normalized Eisenstein series of weights 4 and 6: “normalized” means with constant coefficient equal to 1. We have natural ring homomorphisms, also referred to as δ-Fourier expansion maps [5],

(4.35)  M^{r} → S_{∞}^{r}
g ↦ g_{∞} = g(q, q′, …, q^{(r)}),

characterized by the properties that they send a_{4} and a_{6} to −2^{−4}3^{−1}E_{4}(q) and 2^{−5}3^{−3}E_{6}(q), respectively, and commute with δ. There exists a unique E_{p−1} ∈ M^{0}(p−1) such that E_{p−1}(q) is the normalized Eisenstein series of weight p − 1.

By (4.1) and (7.26) in [5], there exists a unique f^{1} ∈ M^{1}(−1 − ϕ), defined over Z_{p}, such that

(4.36)  f^{1}(q, q′) = \frac{1}{p} log \frac{q^{ϕ}}{q^{p}} := ∑_{n≥1}(−1)^{n−1}n^{−1}p^{n−1} (\frac{q′}{q^{p}} )^{n} ∈ R((q))ˆ [q′]ˆ .

As explained in [5, pp. 126–129], f^{1} is isogeny covariant and may be interpreted as a (characteristic zero) arithmetic Kodaira-Spencer class.

Lemma 4.37. Let E be an elliptic curve given by y^{2} = x^{3} + Ax + B with A, B ∈ R. With notation as above, f^{1}(A, B) = 0 if and only if E is CL.

Proof. See [7, Proposition 7.15]. □

Define

(4.38)  t^{\frac{ϕ+1}{2}} := t^{\frac{p+1}{2}} (\frac{t^{ϕ}}{t^{p}} )^{1/2} = t^{\frac{p+1}{2}} (1 + p\frac{δt}{t^{p}})^{1/2} = t^{\frac{p+1}{2}} ∑_{j≥0} \binom{1/2}{j}p^{j} (\frac{δt}{t^{p}})^{j};

this function is an element of M_{{a_{4}a_{6}}}^{1}(1 + ϕ). Next define

(4.39)  f^{♭} := f^{1} · t^{\frac{ϕ+1}{2}} ∈ M_{\{a_{4}a_{6}\}}^{1}(0) ⊂ M_{\{a_{4}a_{6}g\}}^{1}(0) ⊂ O^{1}(X^{†}).

The maps in (4.34) and (4.35) are compatible, so

(4.40)  f_{∞}^{♭} ∈ q′R((q))ˆ [q′]ˆ ⊂ 𝔮′R((𝔮))ˆ [𝔮′]ˆ .

Finally, by the main theorem of [20],

(4.41)  f^{1} = cE_{p−1}Δ^{−p}(2a_{4}^{p}a′_{6} − 3a_{6}^{p}a′_{4}) + f_{0} + pf_{1},

for some c ∈ R^{×}, f_{0} ∈ M^{0}(−1 − p), and f_{1} ∈ M^{1}. On the other hand, (4.29) implies

b^{ϕ} = (\frac{a_{6}^{2}}{a_{4}^{3}} )^{ϕ} = \frac{(a_{6}^{p} + pa′_{6})^{2}}{(a_{4}^{p} + pa′_{4})^{3}} ,

so a calculation using the definition in (4.9) yields

(4.42)  δb = a_{4}^{−4p}a_{6}^{p}(2a_{4}^{p}a′_{6} − 3a_{6}^{p}a′_{4}) + ph

for some h ∈ M_{{a_{4}a_{6}}}^{1}. Set a_{0} := cE_{p−1}Δ^{−p}a_{4}^{4p}a_{6}^{−p}. Then combining (4.41) and (4.42) yields

(4.43)  f^{♭} = f^{1} · t^{\frac{ϕ+1}{2}} = a_{0}t^{\frac{p+1}{2}} δb + f_{0}t^{\frac{p+1}{2}} + ph_{1},

for some h_{1} ∈ M_{\{a_{4}a_{6}\}}^{1}. Let α = a_{0}t^{\frac{p+1}{2}} ∈ M_{\{a_{4}a_{6}\}}^{0}(0). Then by (4.43) and (4.5), respectively, we obtain, for n = 0 and n = 1,

(4.44)  δ^{n}f^{♭} = α^{p^{n}}δ^{n+1}b + β_{n} + pγ_{n},

for some β_{n} ∈ M_{{a_{4}a_{6}}}^{n}(0) and γ_{n} ∈ M_{{a_{4}a_{6}}}^{n+1}(0).

Lemma 4.45. Assume that the element g ∈ M^{0}(m) is in E_{p−1}M^{0}. Then \overline{f^{♭}} and \overline{δf^{♭}} are algebraically independent over O(X̄^{†}), and the natural maps

(4.46)  O(X̄^{†})[\overline{f^{♭}}] → O^{1}(X^{†}) ⊗_{R} k
(4.47)  O(X̄^{†})[\overline{f^{♭}}, \overline{δf^{♭}}] → O^{2}(X^{†}) ⊗_{R} k
(4.48)  O(X^{†})ˆ → O^{2}(X^{†})/(f^{♭}, δf^{♭})

are isomorphisms.

Proof. By (4.36), (4.38), and (4.39), we have

(4.49)  \overline{f_{∞}^{♭}} = t∞^{\frac{p+1}{2}} q′/q^{p},

which involves q′, so the algebraic independence follows. Reducing (4.31) mod p gives isomorphisms like (4.46) and (4.47) but with \overline{b′} and \overline{b′′} on the left in place of \overline{f^{♭}} and \overline{δf^{♭}}. To change variables, observe that since g ∈ E_{p−1}M^{0}, the element α is invertible in O(X^{†}); thus (4.44) implies O(X̄^{†})[\overline{f^{♭}}] ≃ O(X̄^{†})[\overline{b′}] and O(X̄^{†})[\overline{f^{♭}}, \overline{δf^{♭}}] ≃ O(X̄^{†})[\overline{b′}, \overline{b′′}]. This proves (4.46) and (4.47).

Now (4.47) implies that (4.48) induces an isomorphism mod p, Since both sides of (4.48) are p-adically complete and separated rings, (4.48) is surjective. The δ-Fourier expansion map O^{2}(X^{†}) → R((𝔮))ˆ [𝔮′, 𝔮′′]ˆ followed by the evaluation map mapping 𝔮′ and 𝔮′′ to 0 induces a map O^{2}(X^{†})/(f^{♭}, δf^{♭}) → R((𝔮)), by (4.40). The composition of (4.48) with this is simply the Fourier expansion map, since elements of O(X^{†})ˆ have Fourier expansions in R((𝔮)). So the Fourier expansion principle implies that (4.48) is injective. □

Corollary 4.50. The series \overline{f^{(0)}(q)} and \overline{f_{[a_{p}]}^{(0)}(q)} are Fourier expansions of weight-2 quotients of modular forms.

Proof. We have \overline{f^{(0)}(q)} = (θ^{p−1}f̄) /Ē_{p−1}^{p+1}, which is the Fourier expansion of a weight-2 quotient. We handle the second series in an indirect way, using f^{♯}. Although f^{♯} ∈ O^{2}(X^{†}), we have \overline{f^{♯}} ∈ O^{1}(X^{†}) ⊗_{R} k by (4.23). So (4.46) identifies \overline{f^{♯}} with a polynomial in O(X̄^{†})[\overline{f^{♭}}] ⊂ L[\overline{f^{♭}}], where L := k(\overline{X_{1}(N)}). We can find this polynomial explicitly from the δ-Fourier expansion, since elements of L have expansions in k((q)) while \overline{f_{∞}^{♭}} involves q′: see (4.49). By Lemma 4.20 and (4.49),

\overline{f_{∞}^{♯}} = \begin{cases}\overline{f^{(−1)}(q)} + t_{∞}^{− \frac{p^{2}+p}{2}} (\overline{f_{[a_{p}]}^{(0)}(q)})^{p^{2}} \overline{f_{∞}^{♭}}^{p} − ā_{p}t_{∞}^{− \frac{p+1}{2}} (\overline{f_{[a_{p}]}^{(0)}(q)})^{p} \overline{f_{∞}^{♭}}, & if A is not CL; \\ −ū\overline{f_{[u]}^{(−1)}(q)} + t_{∞}^{− \frac{p+1}{2}} (\overline{f_{[a_{p}]}^{(0)}(q)})^{p} \overline{f_{∞}^{♭}}, & if A is CL. \end{cases}

In either case, taking the coefficient of \overline{f_{∞}^{♭}} shows that ā_{p}t_{∞}^{− \frac{p+1}{2}} (\overline{f_{[a_{p}]}^{(0)}(q)})^{p} is the Fourier

expansion of an element of L. Since t is a weight 2 quotient, ā_{p} (\overline{f_{[a_{p}]}^{(0)}(q)})^{p} is the Fourier expansion of a weight p + 1 quotient, and hence (by dividing by Ē_{p−1}) also of a weight-2 quotient. By (3.31),

−ā_{p} (\overline{f_{[a_{p}]}^{(0)}(q)})^{p} + \overline{f_{[a_{p}]}^{(0)}(q)} = \overline{f^{(0)}(q)};

now \overline{f_{[a_{p}]}^{(0)}(q)} is the Fourier expansion of a weight-2 quotient since the other terms are. □

Remark 4.51. The proof that \overline{f_{[a_{p}]}^{(0)}(q)} is a Fourier expansion of a quotient of modular forms used the theory of δ-modular forms; we know no direct proof.

Recall the Igusa curve I_{1}(N) and its quotient J defined in Section 3.8.

Lemma 4.52. The Fourier series of any modular form f on X_{1}(N) over k is also the Fourier series of a rational function g ∈ k(I_{1}(N)). If the weight of f is even, then we may take g ∈ k(J).

Proof. By [17, Proposition 2.2], there is a line bundle ω on \overline{X_{1}(N)} such that for each i ∈ Z, the global sections of ω^{i} are the modular forms of weight i. We denote by ω also the pullback of ω to I_{1}(N) or J. By [17, p. 461], the sections of ω^{i} on I_{1}(N) or J have naturally defined Fourier expansions, compatible with the Fourier expansions of modular forms on X_{1}(N). There is a section a of ω on I_{1}(N) whose Fourier expansion is 1: see [17, Proposition 5.2]. Given a modular form f of weight i on X_{1}(N), let g := f/a^{i} ∈ k(I_{1}(N)).

The action of F_{p}^{×} on I_{1}(N) lifts to an action of F_{p}^{×} on ω, and −1 ∈ F_{p}^{×} sends a to −a (see [17, Proposition 5.2(5)]), so if i is even, f/a^{i} ∈ k(J). □

Recall the definition of M_{{g}}^{r}(w) from the end of the first paragraph of Section 4.7. By Construction 3.2 and Theorem 5.1 of [1], there exist unique δ-modular forms f^{∂} ∈ M_{{E_{p−1}}}^{1}(ϕ−1) and f_{∂} ∈ M_{{E_{p−1}}}^{1}(1 − ϕ), defined over Z_{p}, with δ-Fourier expansions identically equal to 1. Moreover, these forms are isogeny covariant and f^{∂} · f_{∂} = 1. Furthermore, the reduction \overline{f^{∂}} ∈ M^{1} ⊗ k equals the image of Ē_{p−1} ∈ M_{p−1} in M^{1} ⊗ k. For λ ∈ R^{×}, define

(4.53)  f_{λ} := (f^{1})^{ϕ} − λf^{1}(f^{∂})^{−ϕ−1} ∈ M_{{E_{p−1}}}^{2}(−ϕ − ϕ^{2}).

Since f_{1} and f^{∂} are isogeny covariant, so is f_{λ}. Furthermore consider the series

t^{\frac{ϕ^{2}+ϕ}{2}} := t^{\frac{p^{2}+p}{2}} (\frac{t^{ϕ}}{t^{p}} )^{1/2} (\frac{t^{ϕ^{2}}}{t^{p^{2}}} )^{1/2} ∈ M_{\{a_{4}a_{6}\}}^{2}(ϕ + ϕ^{2}),

and define

(4.54)  f_{λ}^{♭} := f_{λ} · t^{\frac{ϕ^{2}+ϕ}{2}} ∈ M_{\{a_{4}a_{6}E_{p−1}\}}^{2}(0).

The main reason for considering these forms comes from the following

Lemma 4.55. Let E_{1} be an elliptic curve y^{2} = x^{3}+A_{1}x+B_{1} over R with ordinary reduction. Then (1) There exists λ ∈ R^{×} such that f_{λ}(A_{1}, B_{1}) = 0.

(2) If λ is as in (1) and there is an isogeny of degree prime to p between E_{1} and an elliptic curve E_{2} over R given by y^{2} = x^{3} + A_{2}x + B_{2}, then f_{λ}(A_{2}, B_{2}) = 0.

(3) If in addition, A_{2}B_{2} ≢ 0 (mod p), then f_{λ}^{♭}(A_{2}, B_{2}) = (δf_{λ}^{♭})(A_{2}, B_{2}) = ⋯ = 0.

Proof.

(1) If f^{1}(A_{1}, B_{1}) = 0, any λ ∈ R^{×} will do. If f^{1}(A_{1}, B_{1}) ≠ 0, set

λ := \frac{f^{1}(A_{1}, B_{1})^{ϕ}}{f^{1}(A_{1}, B_{1})} f^{∂}(A_{1}, B_{1})^{ϕ+1};

the numerator and denominator of the first factor have the same p-adic valuation and \overline{f^{∂}(A_{1}, B_{1})} ≡ Ē_{p−1}(Ā, B̄) ≠ 0, so λ ∈ R^{×}.

(2) Scaling A_{2} and B_{2} by suitable elements of R^{×}, we may assume that the isogeny pulls back dx/y to dx/y. Now use the isogeny covariance of f_{λ}.

(3) By (4.54), f_{λ}^{♭}(A_{2}, B_{2}) = 0. Now use δ0 = 0.

□

Set σ := q′/q^{p}. Then (4.36), (4.53), and (4.54) yield

(4.56)  \overline{f_{∞}^{1}} = σ,  \overline{f_{λ,∞}} = σ^{p} − \overline{λ}σ,  and \overline{f_{λ,x_{∞}}^{♭}} = t∞^{\frac{p^{2}+p}{2}} (σ^{p} − \overline{λ}σ).

In what follows we assume that X^{†} = U \ [x_{∞}] where U has an étale coordinate τ ∈ O(U) such that [x_{∞}] is scheme-theoretically given by τ: we can arrange this by shrinking X^{†}. Then R[[𝔮]] = R[[τ]], so

R((τ))ˆ [τ′, …, τ^{(r)}]ˆ = R((𝔮))ˆ [𝔮′, …, 𝔮^{(r)}]ˆ = R((𝔮))ˆ [q′, …, q^{(r)}]ˆ .

Also O^{r}(X^{†}) = O(X^{†})ˆ [τ′, …, τ^{(r)}]ˆ . Since

(4.57)  \overline{f_{λ}^{♭}} ∈ O(X̄^{†})[τ′, τ′′] ∩ k((τ))[τ′] = O(X̄^{†})[τ′] = O^{1}(X^{†}) ⊗_{R} k,

we may define a quotient ring

(4.58)  A^{‡} := (O^{1}(X^{†}) ⊗_{R} k)/(\overline{f_{λ}^{♭}})

and a scheme X̄^{‡} := Spec A^{‡}. View A^{‡} as an algebra over A^{†} := O(X^{†}) ⊗ k = O(X̄^{†}).

Lemma 4.59. The k((𝔮))-algebra A^{‡} ⊗_{A^{†}} k((𝔮)) is a product of p copies of k((𝔮)).

Proof. We have

A^{‡} ⊗_{A^{†}} k((𝔮)) = (O(X̄^{†})[τ′]/(\overline{f_{λ}^{♭}})) ⊗_{A^{†}} k((τ))

= k((τ))[τ′]/(\overline{f_{λ}^{♭}})

= k((𝔮))[q′]/(\overline{f_{λ,x_{∞}}^{♭}})

= k((𝔮))[σ]/(σ^{p} − λ̄σ)

≃ ∏_{i=1}^{p} k((𝔮)),

since σ^{p} − λ̄σ = ∏_{i=1}^{p}(σ − λ_{i}) for some λ_{i} ∈ k. Explicitly, the last isomorphism is given by

(4.60)  q′ ↦ (λ_{1}q^{p}, …, λ_{p}q^{p}).

□

Lemma 4.61. One can choose X^{†} so that X̄^{‡} → X̄^{†} is a finite étale cover of degree p.

Proof. By definition, X̄^{‡} → X̄^{†} is of finite type. Lemma 4.59 shows that it is étale of degree p above the generic point of X̄^{†}. Therefore X̄^{‡} → X̄^{†} is finite étale of degree p over some open neighborhood of the generic point. □

In case our correspondence arises from a modular parametrization one has the following variant of Lemma 4.59.

Lemma 4.62. Assume S ⟵^{Π} X ⟶^{Φ} A arises from a modular parametrization and let L = k(\overline{X_{1}(N)}). Then

A^{‡} ⊗_{A^{†}} L ≃ L × A^{+} × A^{−}

where

A^{±} := L[y]/ (y^{(p−1)/2} − Ē_{p−1}/t^{(p−1)/2}) .

Proof. By (4.46), we have A^{‡} ⊗_{A^{†}} L ≃ L[\overline{f^{♭}}]/(\overline{f_{λ}^{♭}}). On the other hand

\overline{f_{λ}^{♭}} = t^{\frac{p^{2}+p}{2}} [(\overline{f^{1}})^{p} − λ̄\overline{f^{1}}(\overline{f^{∂}})^{−p−1}]
= (\overline{f^{♭}})^{p} − λt^{\frac{p^{2}−1}{2}} Ē_{p−1}^{−p−1}\overline{f^{♭}}
= \overline{f^{♭}} [(\overline{f^{♭}})^{(p−1)/2} + \sqrt{λ}t^{(p−1)/2}Ē_{p−1}^{−1}(t^{(p−1)/2}/Ē_{p−1})^{(p−1)/2}]
· [(\overline{f^{♭}})^{(p−1)/2} − \sqrt{λ}t^{(p−1)/2}Ē_{p−1}^{−1}(t^{(p−1)/2}/Ē_{p−1})^{(p−1)/2}] ,

so the result follows. □

4.8. δ-modular forms: Shimura-elliptic case. We continue using the notation and assumptions of Sections 4.5 and 4.6. Assume that the U in (4.11) is small enough that the line bundle of fake 1-forms on U is trivial. (See [7, p. 230] for the definition of this line bundle: there it is called the “line bundle of false 1-forms”.) Let q := 1 + t ∈ R[[t]] and write q′ = δ(1 + t), q′′ = δ^{2}(1 + t), and so on. Define

Ψ = Ψ(t, t′) := \frac{1}{p} log \frac{q^{ϕ}}{q^{p}} = \frac{q′}{q^{p}} − \frac{p}{2} (\frac{q′}{q^{p}} )^{2} + ⋯ ∈ R[[t]][t′]ˆ .

Lemma 4.63. There exists f^{♭} ∈ O^{1}(U) such that

(4.64)  f_{x_{0}}^{♭} = u(t)^{ϕ+1} · Ψ(t, t′) ∈ q′R[[t]][t′]ˆ

for some u(t) ∈ R[[t]]^{×}, and

(4.65)  f^{♭}(P) = 0 for P ∈ Π^{−1}(CL) ∩ U(R).

Proof. Use (8.116), (8.82), and Proposition 8.61 in [7]. (In the notation of [7], one takes f^{♭} to be the value of the “δ-modular form” f_{crys}^{1} at the pull back to U of the universal fake elliptic curve equipped with some invertible fake 1-form; again f_{crys}^{1} should be viewed as an arithmetic Kodaira-Spencer class.) □

Lemma 4.66. There exists a neighborhood X^{†} ⊂ U of the section x_{0} such that \overline{f^{♭}} and \overline{δf^{♭}} are algebraically independent over O(X̄^{†}) and the natural maps

(4.67)  O(X̄^{†})[\overline{f^{♭}}] → O^{1}(X^{†}) ⊗_{R} k
(4.68)  O(X̄^{†})[\overline{f^{♭}}, \overline{δf^{♭}}] → O^{2}(X^{†}) ⊗_{R} k
(4.69)  O(X^{†})ˆ → O^{2}(X^{†})/(f^{♭}, δf^{♭})

are isomorphisms.

Proof. By (4.64),

(4.70)  \overline{f_{x_{0}}^{♭}} = \frac{ū(t)^{p+1}}{(1 + t)^{p}} t′ + S_{0} ∈ k[[t]][t′],

for some S_{0} ∈ k[[t]]. Using (4.5) one obtains

(4.71)  \overline{δf_{x_{0}}^{♭}} = \frac{ū(t)^{p^{2}+p}}{(1 + t)^{p^{2}}} t′′ + S_{1} ∈ k[[t]][t′, t′′]

for some S_{1} ∈ k[[t]][t′]. We may assume that there is an étale coordinate τ on U such that x_{0} is given scheme-theoretically by τ = 0. Then R[[t]] = R[[τ]] (and R[[t]][t′, t′′]ˆ = R[[τ]][τ′, τ′′]ˆ ) so t = S(τ) := ∑_{n≥1} c_{n}τ^{n} for some c_{n} ∈ R with c_{1} ∈ R^{×}. One can easily see that

t′ = \frac{1}{p} [∑ c_{n}^{ϕ}(τ^{p} + pτ′)^{n} − (∑ c_{n}τ^{n})^{p}] = (∂S/∂τ)^{p}τ′ + B_{0} + pB_{1}

for some B_{0} ∈ R[[τ]] and B_{1} ∈ R[[τ]][τ′]ˆ . Using (4.5) we obtain

t′′ = (∂S/∂τ)^{p^{2}}τ′′ + B_{1}^{∗} + pB_{2}

for some B_{1}^{∗} ∈ R[[τ]][τ′]ˆ and B_{2} ∈ R[[τ]][τ′, τ′′]ˆ . Combining with (4.70) and (4.71) and setting

v̄(τ) := \frac{ū(S̄(τ))^{p+1}(∂S̄/∂τ)^{p}}{(S̄(τ) + 1)^{p}} ∈ k[[τ]],

we obtain

(4.72)  \overline{f_{x_{0}}^{♭}} = v̄(τ)τ′ + C_{0}(τ) ∈ k[[τ]][τ′],
\overline{δf_{x_{0}}^{♭}} = v̄(τ)^{p}τ′′ + C_{1}(τ, τ′) ∈ k[[τ]][τ′, τ′′].

where C_{0}(τ) ∈ k[[τ]] and C_{1}(τ, τ′) ∈ k[[τ]][τ′]. On the other hand, by (4.6), we have \overline{f^{♭}} ∈ O(Ū)[τ′] and \overline{δf^{♭}} ∈ O(Ū)[τ′, τ′′]. Thus v̄(τ), C_{0}(τ), and C_{1}(τ, τ′) are images of elements v̄ ∈ O(Ū), C_{0} ∈ O(Ū), and C_{1} ∈ O(Ū)[τ′], respectively, such that

(4.73)  \overline{f^{♭}} = v̄τ′ + C_{0}, and \overline{δf^{♭}} = v̄^{p}τ′′ + C_{1}.

Lift v̄ to v ∈ O(U). Let X^{†} be the complement in U of the closed subscheme defined by v. Since v̄(τ) has a nonzero constant term, v̄ does not vanish at x̄_{0}, so X^{†} contains the section x_{0}. The proof now follows the proof of Lemma 4.45, using (4.73) in place of (4.44). □

Remark 4.74. Using [7, pp. 268–269], for Ū contained in the ordinary locus one can construct forms f_{λ}^{♭} ∈ O^{2}(U) analogous to the ones in (4.54). (In the notation of [7], one takes f_{λ}^{♭} to be the Shimura analogues of the forms in (4.53) evaluated at a basis of the module of fake 1-forms on U.) The analogues of Lemmas 4.55, 4.59, and 4.61 still hold with Fourier expansions replaced by Serre-Tate expansions. The corresponding statements and their proofs are analogous to the ones in the modular-elliptic case.

4.9. Proofs of the local results.

Proof of Theorem 3.5. Assume that we are given either a modular-elliptic or a Shimuraelliptic correspondence S ⟵^{Π} X ⟶^{Φ} A. Assume that p is sufficiently large and p splits completely in F_{0}. In the Shimura-elliptic case we also assume that the places v|p are not anomalous for A. In the modular-elliptic case, choose g as in Lemma 4.45 and define X^{†} as in (4.30). In the Shimura-elliptic case, choose X^{†} as in Lemma 4.66. By Lemma 4.45 or 4.66, there exists Φ^{†} ∈ O(X^{†})ˆ such that

(4.75)  f^{♯} − Φ^{†} = h_{0}f^{♭} + h_{1}δf^{♭},

for some h_{j} ∈ O^{2}(X^{†}). Suppose that P_{1}, …, P_{n} ∈ Π^{−1}(CL) ∩ X^{†}(R) and m_{1}, …, m_{n} ∈ Z. By Lemma 4.37 or 4.63, we have f^{♭}(P_{i}) = 0, so δf^{♭}(P_{i}) = 0. Thus

(4.76)  f^{♯}(P_{i}) = Φ^{†}(P_{i}).

Now (4.76) implies

(4.77)  ∑ m_{i}Φ^{†}(P_{i}) = ∑ m_{i}f^{♯}(P_{i}) = ∑ m_{i}ψ(Φ(P_{i})) = ψ (∑ m_{i}Φ(P_{i})) .

Equation (4.77) and Lemma 4.7(4) imply the second of the two equivalences in Theorem 3.5.

We now prove the first equivalence in Theorem 3.5. Let Q := ∑ m_{i}Φ(P_{i}). If Q ∈ A(R)_{tors}, then ψ(Q) = 0 and (4.77) implies ∑ m_{i}Φ^{†}(P_{i}) = 0. Conversely, suppose that ∑ m_{i}Φ^{†}(P_{i}) = 0; then Q ∈ ker ψ. Since CL ⊆ S(\overline{Q}), we have P_{i} ∈ X(\overline{Q}) ∩ X(R), so Q ∈ A(\overline{Q}) ∩ A(R) ⊂ A(Z_{p}^{ur}). So Lemma 4.7(5) implies Q ∈ A(R)_{tors}.

To complete our proof, we need to check that \overline{Φ^{†}} ∉ k.

Assume first that we are in the modular-elliptic case. By (4.40),

(4.78)  δf_{∞}^{♭} ∈ (𝔮′, 𝔮′′)R((𝔮))ˆ [𝔮′, 𝔮′′]ˆ .

Taking δ-Fourier expansions in (4.75), taking ♮ (i.e., setting 𝔮′ = 𝔮′′ = 0), and using (4.40) and (4.78), we obtain

(4.79)  Φ_{x_{∞}}^{†} = (f_{x_{∞}}^{♯})_{♮}.

Let e be the ramification index of Φ: X → A at x_{∞}. Then the b_{e} ∈ F_{0} of Section 4.6.1 is nonzero. We may assume that p is large enough that e, b_{e} ≢ 0 (mod p). By (4.79) and (4.15), the coefficient of 𝔮^{e} in Φ_{x_{∞}}^{†} is \frac{b_{e}}{e} or −u\frac{b_{e}}{e}, where u ≢ 0 (mod p); in either case this coefficient

is nonzero mod p. Thus \overline{Φ_{x_{∞}}^{†}} ∉ k. Hence \overline{Φ^{†}} ∉ k.

Finally, assume that we are in the Shimura-elliptic case. By (4.64),

(4.80)  δf_{x_{0}}^{♭} ∈ (q′, q′′)R[[t]][t′, t′′]ˆ .

By (4.5),

(4.81)  \begin{aligned}q′ & = & t′ − G_{1}(t) \\ q′′ & = & t′′ − G_{2}(t, t′), \end{aligned}

for some G_{1}(t) ∈ Z[t] and G_{2}(t, t′) ∈ Z[t, t′]. Denote by G ↦ G_{♮} the substitution homomorphism

R[[t]][t′, t′′]ˆ → R[[t]]

sending t′ to G_{1}(t) and t′′ to G_{2}(t, t′). Then (q′)_{♮} = (q′′)_{♮} = 0, so (4.64) and (4.80) imply (f_{x_{0}}^{♭})_{♮} = (δf_{x_{0}}^{♭})_{♮} = 0. Taking δ-Serre-Tate expansions in (4.75), taking ♮, and substituting (4.25), we obtain

(4.82) Φ_{x_{0}}^{†} = \begin{cases}ψ(z_{0}) + \frac{1}{p} ∑_{n≥1} (\frac{b_{n}^{ϕ^{2}}}{n} ((t^{ϕ^{2}})_{♮})^{n} − a_{p} \frac{b_{n}^{ϕ}}{n} ((t^{ϕ})_{♮})^{n} + p\frac{b_{n}}{n} t^{n}) , & if A is not CL; \\ ψ(z_{0}) + \frac{1}{p} ∑_{n≥1} (\frac{b_{n}^{ϕ}}{n} ((t^{ϕ})_{♮})^{n} − up\frac{b_{n}}{n} t^{n}) , & if A is CL. \end{cases}

Substituting the two formulas

(t^{ϕ})_{♮} = (q^{ϕ} − 1)_{♮} = (q^{p} + pq′ − 1)_{♮} = q^{p} − 1 = (1 + t)^{p} − 1 = pt + ⋯ + t^{p}, and
(t^{ϕ^{2}})_{♮} = (q^{ϕ^{2}} − 1)_{♮} = ((q^{p} + pq′)^{p} + p((q′)^{p} + pq′′) − 1)_{♮} = q^{p^{2}} − 1 = (1 + t)^{p^{2}} − 1 = p^{2}t + ⋯ + t^{p^{2}},

and recalling from Section 4.6.2 that b_{1} = 1, we deduce that the coefficient of t in Φ_{x_{0}}^{†} is 1 − a_{p} + p if A_{R} is not CL, and 1 − u if A_{R} is CL. This coefficient is nonzero mod p, since our non-anomalous assumption implies a_{p} ≢ 1 (mod p) and we have ūā_{p} = 1 in the CL case. So Φ_{x_{0}}^{†} ∉ R + pR[[t]]. Hence \overline{Φ^{†}} ∉ k. □

Proof of Theorem 3.17. Assume, in the proof of Theorem 3.5, that our modular-elliptic correspondence S ⟵^{Π} X ⟶^{Φ} A satisfies S = X = X_{1}(N), Π = Id, and Φ is a modular parametrization attached to a newform f. We may choose g := E_{p−1} in Section 4.7; then X̄^{†} = \overline{Y_{1}(N)}^{ord} \ \{x | j(x) = 0, 1728\}. Now (4.19) and (4.79) give the formula for Φ_{∞}^{†}. □

The following lemma was needed to prove Theorem 3.24; the notation is as in Theorem 3.24.

Lemma 4.83. Suppose that \overline{Θ} ∈ F. Assume that for every P ∈ P and every prime l ≠ p we have

(4.84)  ∑_{i} \overline{Θ}(P̄_{i}^{(l)}) − a_{l}\overline{Θ}(P̄) = 0.

Then \overline{Θ} = λ̄\overline{f^{(−1)}} for some λ̄ ∈ k.

Proof. Since \overline{Θ} is regular on \overline{X_{1}(N)}^{ord}, there exists m ∈ Z_{≥1} such that Ḡ := Ē_{p−1}^{m}\overline{Θ} is a modular form over k on Γ_{1}(N). View modular forms as functions on the set of triples (E, α, ω) where E is an elliptic curve over k, where α: Z/nZ ↪ E(k) is an injective homomorphism, and ω is a nonzero 1-form on E. Given P ∈ P and a prime l ≠ p, choose (E, α, ω) such that (E, α) represents P, and choose (E_{i}, α_{i}, ω_{i}) such that (E_{i}, α_{i}) represents P_{i}^{(l)} and such that ω_{i} pulls back to ω under the l-isogeny E → E_{i}. Then Ē_{p−1}(E_{i}, α_{i}, ω_{i}) = Ē_{p−1}(E, α, ω) by [7, p. 269], for instance. Multiplying (4.84) by this yields

∑_{i} Ḡ(E_{i}, α_{i}, ω_{i}) = a_{l}Ḡ(E, α, ω).

By [17, p. 452] or [21, p. 90], the left hand side equals (lT (l)Ḡ)(E, α, ω). Since P is infinite, it follows that lT (l)Ḡ = a_{l}Ḡ for all l ≠ p. On the other hand, Ḡ(q) = \overline{Θ}(q), and U\overline{Θ} = 0, so UḠ = 0. Furthermore Ḡ is invariant under the diamond operators. Thus Ḡ is a Hecke eigenform with the same eigenvalues as θ^{p−2}f̄, so by [17, p. 453], we have Ḡ(q) = λ̄·(θ^{p−2}f̄)(q) for some λ̄ ∈ k. Thus \overline{Θ}(q) = λ̄\overline{f^{(−1)}}(q), so \overline{Θ} = λ̄\overline{f^{(−1)}}. □

Proof of Theorem 3.25. Assume that we have a modular-elliptic correspondence. Pick Q ∈ C represented by (E_{Q}, α_{Q}) where E_{Q} is given by y^{2} = x^{3} + Ax + B. By Lemma 4.55(1), there exists λ ∈ R^{×} such that f_{λ}(A, B) = 0. Let X^{†} ⊂ X satisfy the conclusions of Lemmas 4.45 and 4.61. View f_{λ}^{♭} and f^{♯} as elements of O^{2}(X^{†}); then \overline{f_{λ}^{♭}}, \overline{f^{♯}} ∈ O^{1}(X^{†}) ⊗_{R} k by (4.57) and (4.23), respectively. Let \overline{Φ^{‡}} be the image of \overline{f^{♯}} in the ring A^{‡} = O(X̄^{‡}) of (4.58).

Claim. \overline{Φ^{‡}} is non-constant on each irreducible component of X̄^{‡}.

If not, there is a minimal prime P of A^{‡} such that the image of \overline{Φ^{‡}} in A^{‡}/P, and hence in (A^{‡}/P) ⊗_{A^{†}} k((𝔮)), is in k. By Lemma 4.59, (A^{‡}/P) ⊗_{A^{†}} k((𝔮)) is a nonzero product of copies of k((𝔮)). By (4.60), the element

\overline{f_{x_{∞}}^{♯}} ∈ k[[𝔮]][𝔮′] ⊂ k((𝔮))[q′]

is sent into an element of k by at least one of the k((𝔮))-algebra homomorphisms

(4.85)  k((𝔮))[q′] → k((𝔮)),

denoted s ↦ s_{∗} and defined by (q′)_{∗} := λ_{i}q^{p}. Since q = 𝔮^{M}, we have

q′ = δ(𝔮^{M}) = \frac{(𝔮^{p} + p𝔮′)^{M} − 𝔮^{pM}}{p} ≡ M𝔮^{p(M−1)}𝔮′ (mod p),

so 𝔮′ ≡ M^{−1}𝔮^{−p(M−1)}q′ (mod p). Thus

(𝔮′)_{∗} = M^{−1}𝔮^{−p(M−1)}λ_{i}q^{p} = M^{−1}λ_{i}𝔮^{p} ∈ 𝔮^{p}k[[𝔮]].

Hence

(\overline{f_{x_{∞}}^{♯}})_{∗} ∈ (\overline{f_{x_{∞}}^{♯}})_{♮} + 𝔮^{p}k[[𝔮]],

where we recall that ♮ means setting 𝔮′ = 0. Let e be the ramification index of Φ at x_{∞}. Exactly as in the proof of Theorem 3.5, since p ≫ 0, the coefficient of 𝔮^{e} in (\overline{f_{x_{∞}}^{♯}})_{♮} is nonzero. So (\overline{f_{x_{∞}}^{♯}})_{∗} is not in k, a contradiction. This ends the proof of our Claim.

Now consider the set C := Π^{−1}(C) ∩ X^{†}(R) and let P_{1} ∈ C, Q_{1} := Π(P_{1}). Let E_{Q_{1}} be given by y^{2} = x^{3} + A_{1}x + B_{1}. By choice of X^{†}, we have A_{1}B_{1} ≢ 0 (mod p). By Lemma 4.55, f_{λ}^{♭}(P_{1}) = 0. Therefore the homomorphism O^{1}(X^{†}) → R sending a function to its value at P_{1} induces a homomorphism A^{‡} → k, which may be viewed as a point σ(P_{1}) ∈ X̄^{‡}(k) mapping to P_{1} ∈ X̄^{†}(k). This defines σ: C →→ X̄^{‡}(k). By definition of σ(P_{1}) and Φ^{‡}, \overline{f^{♯}(P_{1})} = \overline{Φ^{‡}}(σ(P_{1})). Now, for P_{1}, …, P_{n} ∈ C,

∑_{i=1}^{n} m_{i}\overline{Φ^{‡}}(σ(P_{i})) = ∑_{i=1}^{n} m_{i}\overline{f^{♯}(P_{i})}

= ∑_{i=1}^{n} m_{i}\overline{ψ(Φ(P_{i}))}

= \overline{ψ (∑_{i=1}^{n} m_{i}Φ(P_{i}))},

so the desired equivalence follows from Lemma 4.7(4).

The case of Shimura-elliptic correspondences is entirely similar, given Remark 4.74. We skip the details but point out one slight difference in the computations. The proof of the analogue of the Claim above, uses a k[[t]]-algebra homomorphism

k[[t]][t′] → k[[t]],

denoted s ↦ s_{∗}, defined by requiring (q′)_{∗} = λ_{i}q^{q}, where q = 1 + t and q′ = δ(1 + t). Then one must check that for f_{x_{0}}^{♯} as in (4.25), the coefficient of t in (f_{x_{0}}^{♯})_{∗} is nonzero mod p. This coefficient can be computed explicitly, and, unlike in the modular-elliptic case, its expression has contributions from all the terms with n ≥ 1. Nevertheless all the contributions from terms with n ≥ 2 turn out to be 0 mod p, and the coefficient in question turns out to be congruent mod p to either 1 − a_{p} or 1 − u, and hence is nonzero mod p. □

The following will be used to prove Theorem 1.6:

Lemma 4.86. Under the assumptions of Theorem 1.6 there is a constant γ depending only on N such that all the fibers of the reduction mod p map C → \overline{C} are finite of cardinality at most γ.

Proof. Assume that we are in the modular-elliptic case; the Shimura-elliptic case follows by the same argument. Suppose that Q_{1}, Q_{2} ∈ C are such that Q̄_{1} = Q̄_{2} ∈ S(k). Let Q_{i} be represented by (E_{i}, α_{i}), so there is an isogeny u: E_{1} → E_{2} of degree ∏ l_{j}^{e_{j}} where the l_{j} are inert in K_{Q}.

We claim that E_{1} ≃ E_{2}. Since Ē_{1} ≃ Ē_{2}, we may view ū as an element of End Ē_{1}, which may be identified with a subring of the ring of integers O of K_{Q}. The norm of this element equals deg ū = deg u, but the only elements of O whose norm is a product of inert primes are those in Z · O^{×}. Hence u factors as E_{1} →^{n} E_{1} →^{ϵ} E_{2} for some n ∈ Z and ϵ of degree 1. In particular, E_{1} ≃ E_{2}.

By the claim, Lemma 4.86 holds with γ equal to the number of possible Γ_{1}(N)-structures on an elliptic curve. □

Proof of Theorem 1.6. By Lemma 4.86, the map

Φ(Π^{−1}(C)) ∩ Γ → \overline{Φ(Π^{−1}(C)) ∩ Γ}

has finite fibers of cardinality bounded by a constant independent of Γ. On the other hand, by Corollary 3.29, the target of this map has cardinality at most cp^{r} for some c independent of Γ. □

Proof of Theorem 3.32. Assume, in the proof of Theorem 3.25, that we have a modularelliptic correspondence arising from a modular parametrization attached to f. Part (1) follows from Lemma 4.62. Part (2) follows comparing Fourier expansions of the two sides: apply the substitution maps as in (4.85) to \overline{f_{∞}^{♯}} given in (4.14) to obtain the p different series

\overline{Φ^{‡}}_{∞i} = (\overline{f_{∞}^{♯}})_{∗} = \overline{Φ^{†}}(q) + \overline{λ}_{i}\overline{Φ^{††}}(q) ∈ k((q))

where λ_{1}, …, λ_{p} ∈ k are the zeros of x^{p} − λ̄x as in the proof of Lemma 4.59. □

Appendix A. Non-existence of reciprocity in the global case

Let S ⟵^{Π} X ⟶^{Φ} A be a modular-elliptic or a Shimura-elliptic correspondence and let X^{†} ⊂ X be a dense open subscheme. Ideally we would like a description of the group of divisors ∑ m_{i}P_{i} on X^{†} supported in Π^{−1}(CM) such that ∑ m_{i}Φ(P_{i}) ∈ A(\overline{Q})_{tors}. More precisely, in analogy with the “local” result Theorem 3.5, one may ask if there exists a regular function Φ^{†} on X^{†} such that for any divisor ∑ m_{i}P_{i} on X^{†} supported in Π^{−1}(CM) we have that ∑ m_{i}Φ(P_{i}) ∈ A(\overline{Q})_{tors} if and only if ∑ m_{i}Φ^{†}(P_{i}) = 0. We could refer to such a Φ^{†} as a reciprocity function for CM points.

A.1. Non-existence of global reciprocity functions for isogeny classes and CM points. But even in the “most classical” case of modular-elliptic correspondences arising from a newform, no such function exists:

Theorem A.1 (Non-existence of reciprocity functions for CM points). Let Φ : X_{1}(N) → A be a modular parametrization. Assume that there is a non-empty open subscheme X^{†} ⊂ X_{1}(N) and a regular function Φ^{†} ∈ O(X^{†}) having the property that for any P_{1}, …, P_{n} ∈ CM ∩X^{†}(\overline{Q}) and any m_{1}, …, m_{n} ∈ Z we have

∑_{i=1}^{n} m_{i}Φ(P_{i}) ∈ A(\overline{Q})_{tors} ⇒ ∑_{i=1}^{n} m_{i}Φ^{†}(P_{i}) = 0 ∈ \overline{Q}.

Then Φ^{†} = 0.

Theorem A.1 follows immediately from the following isogeny class analogue applied to an isogeny class of CM points.

Theorem A.2 (Non-existence of reciprocity functions for isogeny classes). Let Φ: S = X_{1}(N) → A be a modular parametrization. Let C ⊂ S(\overline{Q}) be an isogeny class and let Φ^{†} be a rational function on S none of whose poles is in C. Assume that for any P_{1}, …, P_{n} ∈ C and any m_{1}, …, m_{n} ∈ Z we have

(A.3)  ∑_{i=1}^{n} m_{i}Φ(P_{i}) ∈ A(\overline{Q})_{tors} ⇒ ∑_{i=1}^{n} m_{i}Φ^{†}(P_{i}) = 0 ∈ \overline{Q}.

Then Φ^{†} = 0.

Proof. We use Hecke correspondence notation as in Section 3.2. Extend Φ linearly to a homomorphism Φ_{∗}: Div^{0}(X_{1}(N)(\overline{Q})) → A(\overline{Q}). Then Φ_{∗} ∘ T (l)_{∗} = a_{l} · Φ_{∗}; see [14, p. 242].

For any point P ∈ C we have T (l)_{∗}(P − ∞) = ∑ P_{i}^{(l)} − ∑ P_{i0}^{(l)} with P_{i0}^{(l)} cusps. We get

(A.4)  \begin{aligned}a_{l} · Φ(P) & = & a_{l}(Φ_{∗}(P − ∞)) \\ & = & Φ_{∗}(T (l)_{∗}(P − ∞)) \\ & = & Φ_{∗}(∑ P_{i}^{(l)} − ∑ P_{i0}^{(l)}) \\ & = & ∑ Φ(P_{i}^{(l)}) − ∑ Φ(P_{i0}^{(l)}) \end{aligned}

By the Manin-Drinfeld theorem (see [23, p. 62], for instance), Φ(P_{i0}^{(l)}) ∈ A(\overline{Q})_{tors}, so (A.4) yields

(A.5)  ∑ Φ(P_{i}^{(l)}) − a_{l} · Φ(P) ∈ A(\overline{Q})_{tors}.

By (A.3), we obtain ∑_{i} Φ^{†}(P_{i}^{(l)})−a_{l} ·Φ^{†}(P) = 0. Now Lemma A.6 below implies Φ^{†} = 0. □

Lemma A.6. Let S = X_{1}(N). Let f = ∑ a_{n}q^{n} be a weight-2 newform on Γ_{1}(N). let C ⊂ S(\overline{Q}) be an isogeny class. Let Φ^{†} be a rational function on S none of whose poles are in C. Assume that for infinitely many primes l and for any P ∈ C we have

(A.7)  ∑_{i} Φ^{†}(P_{i}^{(l)}) − a_{l}Φ^{†}(P) = 0.

Then Φ^{†} = 0.

Proof. Assume that Φ^{†} ≠ 0. The function

(T (l)Φ^{†})(x) := ∑ Φ^{†}(x_{i}^{(l)}),

defined for all but finitely many x ∈ S(C), is a rational function on S by [39, p. 55]. For the infinitely many given l, the rational functions T (l)Φ^{†} and Φ^{†} agree on the infinite set C so they coincide. Since Φ^{†} may be viewed as a ratio of modular forms over \overline{Q}, each of which is a \overline{Q}-linear combination of newforms whose Fourier coefficients are algebraic integers, the Fourier expansion φ(q) of Φ^{†} is in O_{K,S}((q)) for some ring of S-integers in some number field K, with S finite. We may restrict attention to primes l ∤ N not lying under any prime in S. We may assume also that the leading coefficient of φ(q) is prime to l. The q-values corresponding to the elliptic curves l-isogenous to the one corresponding to q itself are q^{l} and the l-th roots of q, so taking Fourier expansions in T (l)Φ^{†} = Φ^{†} yields

(A.8)  φ(q^{l}) + ∑_{b=0}^{l−1} φ(ζ^{b}q^{1/l}) = a_{l}φ(q),

where ζ is a primitive l-th root of 1. Let v_{q} be the valuation on \overline{Q}((q)). Comparing leading terms in (A.8) yields v_{q}(φ) ≥ 0; and if v_{q}(φ) = 0, then l + 1 = a_{l}, which contradicts |a_{l}| ≤ 2\sqrt{l} < l + 1. Thus v_{q}(φ) > 0.

The series ∑_{b=0}^{l−1} φ(ζ^{b}q^{1/l}) is divisible by l, so

(A.9)  φ(q^{l}) ≡ a_{l}φ(q) (mod lO_{K,S}[[q]].)

The leading coefficient of φ(q^{l}) equals that of φ(q), so it is prime to l. Then (A.9) shows that a_{l} is prime to l. Now (A.9) contradicts v_{q}(φ) > 0. □

A.2. Non-existence of geometric reciprocity functions. Finally we prove that there are no purely geometric reasons for the existence of reciprocity functions; thus the existence of reciprocity functions in the local setting is a truly arithmetic phenomenon.

Theorem A.10 (Non-existence of geometric reciprocity functions). Let Φ: X → A be a non-constant morphism between smooth projective curves over an algebraically closed field k of characteristic p ≥ 0, where A is an elliptic curve. Let n ≥ 3, and let a_{1}, …, a_{n} be nonzero integers not all divisible by p. Suppose that X^{†} ⊂ X is an affine open subset and Φ^{†} ∈ O(X^{†}) is a regular function such that for any P_{1}, …, P_{n} ∈ X^{†}(k) we have

(A.11)  ∑_{i=1}^{n} a_{i}Φ(P_{i}) = 0 ⟹ ∑_{i=1}^{n} a_{i}Φ^{†}(P_{i}) = 0.

Then Φ^{†} is constant. In particular, if ∑_{i=1}^{n} a_{i} is not divisible by p, then Φ^{†} = 0.

Remark A.12. Theorem A.10 fails for both n = 1 and n = 2. Let A be any elliptic curve over k, let X = A, let Φ be the identity, and let X^{†} ⊊ X any nonempty affine open subset. For n = 1, one obtains a counterexample by taking a_{1} = 1 and Φ^{†} ∈ O(X^{†}) any non-constant regular function vanishing at the origin (if the origin is in X^{†}). For n = 2, one obtains a counterexample by taking a_{1} = 1, a_{2} = −1, and Φ^{†} ∈ O(X^{†}) any non-constant regular function. Alternatively, for n = 2, one can take a_{1} = a_{2} = 1 and Φ^{†} a nonconstant rational function that is anti-invariant for the negation map on X (shrink X^{†} if necessary), such as the y-coordinate on a short Weierstrass model in characteristic not 2; this shows that the final sentence of Theorem A.10 can fail too.

Proof of Theorem A.10. Without loss of generality, p ∤ a_{1}. To prove that Φ^{†} is constant, it will suffice to show that Φ^{†} is regular at every P ∈ X(k).

Fix P. Let Y be the inverse image of {0} under the morphism

β: X × (X^{†})^{n−1} → A
(P_{1}, …, P_{n}) ↦ ∑ a_{i}Φ(P_{i}).

Let π_{i}: Y → X be the i-th projection. The morphism π_{1} : Y → X is surjective since given P_{1}, if we choose P_{4}, …, P_{n} ∈ X^{†}(k) arbitrarily, then there are only finitely many choices of P_{2} ∈ X^{†} such that the equation β(P_{1}, …, P_{n}) = 0 forces P_{3} ∉ X^{†}. In particular, we can find a smooth irreducible curve C and a morphism γ: C → Y such that π_{1}(γ(C)) is a dense subset of X containing P.

By (A.11), we have ∑ a_{i}Φ^{†}(P_{i}) = 0 for all (P_{1}, …, P_{n}) ∈ Y ∩ (X^{†})^{n}. In particular,

∑_{i=1}^{n} a_{i}Φ^{†}(π_{i}(γ(c))) = 0

is an identity of rational functions of c ∈ C. Since Φ^{†} is regular on X^{†}, the last n − 1 summands are regular on C. Therefore the first summand is regular too. So a_{1}Φ^{†} is regular on π_{1}(γ(C)). Since a_{1} ≠ 0 in k, and P ∈ π_{1}(γ(C)), the function Φ^{†} is regular at P. □

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University of New Mexico, Albuquerque, NM 87131

  E-mail address: buium@math.unm.edu
  URL: http://math.unm.edu/~buium

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA

  E-mail address: poonen@math.mit.edu
  URL: http://math.mit.edu/~poonen