Mazur’s Conjecture on higher Heegner points

Christophe Cornut

October 24, 2001

Abstract In this article, we establish a non-triviality statement for Heegner points which was conjectured by B. Mazur [10], and has subsequently been used as a working hypothesis by a few authors in the study of the arithmetic of elliptic curves.

Introduction

Let 𝔼_{/ℚ} be an elliptic curve and π : X_{0}(N)_{/ℚ} → 𝔼_{/ℚ} a modular parametrisation. Let K ⊂ ℂ be an imaginary quadratic field, O_{K} its ring of integers and d_{K} its discriminant. Assume the following Heegner Hypothesis: all prime factors of N split in K. Choose an ideal N of O_{K} such that O_{K}/N ≃ ℤ/Nℤ. Then the complex tori ℂ/O_{K} and ℂ/N^{−1} define elliptic curves related by a cyclic N-isogeny, i.e., a complex point x_{1} of X_{0}(N). More generally, if c is a positive integer prime to N, let O_{c} = ℤ + cO_{K} be the order of conductor c in K, put N_{c} = O_{c} ∩ N, and define:

x_{c} = [ℂ/O_{c} → ℂ/N_{c}^{−1}] ∈ X_{0}(N)

The theory of complex multiplication shows that this Heegner point is rational over K[c], the ring class field of conductor c of K — see [4] for the basic properties of these fields. Put y_{c} = π(x_{c}) ∈ 𝔼(K[c]).

The Heegner Hypothesis implies that the L-function of 𝔼_{/K} vanishes to odd order at s = 1. B.H. Gross and D. Zagier [6] established a formula which relates the value of its derivative at 1 to the canonical height of the point Tr_{K[1]/K}(y_{1}) in 𝔼(K), thus proving in particular that:

Tr_{K[1]/K}(y_{1}) ∉ 𝔼(K)_{tors} ⟺ L′(𝔼/K, 1) ≠ 0

In a subsequent work, V.A. Kolyvagin [8] showed the implication:

Tr_{K[1]/K}(y_{1}) ∉ 𝔼(K)_{tors} ⟹ rank(𝔼(K)) = 1

Kolyvagin’s method (together with the input from the Gross-Zagier formula) actually yields almost the complete Conjecture of Birch and Swinnerton-Dyer, provided that the analytic rank ord_{s=1}L(𝔼/K, s) equals 1.

In the general case, B. Mazur suggested that higher Heegner points may still have their word to say about the arithmetic of 𝔼_{/K}. More precisely, let p ∤ N be a prime number^{1}. Then K[p^{∞}] = ⋃_{n≥0} K[p^{n}] is a finite extension of the anticyclotomic ℤ_{p}-extension H_{∞} of K, and Mazur’s conjecture [10], which we will prove here, is the following:

Theorem. There exists n ≥ 0 such that: Tr_{K[p^{∞}]/H_{∞}}(y_{p^{n}}) ∉ 𝔼(H_{∞})_{tors}.

Let us quote some known consequences. Besides extra technical conditions – see the references for further details – all of these corollaries assume that 𝔼_{/ℚ} has (good) ordinary reduction at p.

The theorem signifies that Heegner points yield non-trivial input for the Iwasawa theory of the elliptic curve 𝔼 along H_{∞}/K. Extending Kolyvagin’s method to this Iwasawa setting, M. Bertolini [1] has proved:

• Put Γ = Gal(H_{∞}/K) and Λ = ℤ_{p}[[Γ]]. Let H_{n}/K be the fixed field of Γ^{p^{n}}. Then the Pontryagin dual of Sel_{p^{∞}}(𝔼/H_{∞}) = \underrightarrow{lim}(Sel_{p^{∞}}(𝔼/H_{n})) is a Λ-module of rank one.

The information thus obtained partially descends to K by control theorems due to Mazur. In this direction, J. Nekovář and N. Schappacher [11] have shown:

• The Selmer group Sel_{p^{∞}}(𝔼/K) = \underrightarrow{lim} Sel_{p^{n}}(𝔼/K) contains a copy of ℚ_{p}/ℤ_{p}. In other words: corank_{ℤ_{p}}(Sel_{p^{∞}}(𝔼/K)) ≥ 1. If the p-part of the Tate-Šafarevič group of 𝔼_{/K} is finite, then rank(𝔼(K)) ≥ 1.

Using his theory of Selmer Complexes, J. Nekovář has recently [12] obtained:

• For any elliptic curve 𝔼_{/ℚ},

ord_{s=1}L(𝔼, ℚ) ≡ corank_{ℤ_{p}} (Sel_{p^{∞}}(𝔼/ℚ)) (mod 2)

A central ingredient of our proof is taken from V. Vatsal who, in a preprint of Spring 2000 [19], used a theorem of M. Ratner to deduce an equidistribution statement for Gross points on the connected components of a “definite Shimura curve” (see [5]). In the meantime, Vatsal has also derived a proof of Mazur’s conjecture, under the extra assumption that d_{K} is prime and not equal to p. He uses Jochnowitz congruences to relate his previous results to the setting of classical Heegner points on modular curves [20]. Our more direct and geometrical approach enables us, on the one hand, to relax the assumption on d_{K}, and on the other to generalize Mazur’s conjecture in various ways – see below and section 4 for precise statements. Our analog of Vatsal’s equidistribution result, (Theorem 3.1) may also be of independent interest.

Our proof starts from the simple (but crucial) observation that the torsion subgroup of 𝔼(K[∞]) is finite, where K[∞] = ⋃_{c≥1} K[c] (Lemma 4.1). It follows for instance that almost all points y_{c} ∈ 𝔼(K[∞]) (c ≥ 1) have infinite order, since only a finite number of the distinct points x_{c} ∈ X_{0}(N)(K[∞]) can map to the finite set 𝔼(K[∞])_{tors}. In the same way, Mazur’s conjecture would be proven once we knew that the set {Tr_{K[p^{∞}]/H_{∞}}(y_{p^{n}}) | n ≥ 0} is infinite. This would obviously be true if the set

1See section 5 below for the case where p divides N.

{(σx_{p^{n}})_{σ∈G_{0}} | n ≥ 0} ⊂ X_{0}(N)^{G_{0}}

were Zariski dense, where we put G_{0} = Gal(K[p^{∞}]/H_{∞}).

This is however not the case. Indeed, let Q be a ramified prime of K/ℚ with residue characteristic q ≠ p, and put σ = Frob_{Q}(K[p^{∞}]/K) ∈ Gal(K[p^{∞}]/K). The relation Q^{2} = qO_{K} implies that σ^{2} = 1, hence σ ∈ G_{0} since G_{0} is precisely the torsion subgroup of Gal(K[p^{∞}]/K). But if σ ≠ 1, the Zariski closure of the set {(x_{p^{n}}, σx_{p^{n}}) | n ≥ 0} in X_{0}(N) × X_{0}(N) is a curve, namely the image of the modular curve X_{0}(Nq) under the product of the two classical degeneracy maps X_{0}(Nq) → X_{0}(N) (cf. section 4.1).

In view of this obstruction, we are led to consider the “genus” subgroup:

G_{1} = ⟨ Frob_{Q}(K[p^{∞}]/K) | Q^{2} = qO_{K}, q | d_{K}, q ≠ p⟩ ⊂ G_{0}.

The Galois action of G_{1} on the Heegner points is of geometric nature. More precisely, let M be the product of all primes q | d_{K}, q ≠ p. Then we construct (section 4.1) a family of points (x′_{p^{n}})_{n≥0} ∈ X_{0}(NM)(K[p^{∞}]) and a non-constant morphism π′ : X_{0}(NM) → 𝔼 defined over ℚ such that:

∀n ≥ 0 : π′(x′_{p^{n}}) = Tr_{G_{1}}(y_{p^{n}}) ∈ 𝔼(K[p^{∞}]).

Note that the Heegner Hypothesis does not hold for NM and K.

Choosing a complete set of representatives R ⊂ G_{0} of G_{0}/G_{1}, we now want the set { ∑_{σ∈R} σ · π′(x′_{p^{n}}) | n ≥ 0} to be infinite, and this is again related to the fact that

{(σx′_{p^{n}})_{σ∈R} | n ≥ 0} ⊂ X_{0}(NM)^{R}

is very large. In this direction, a sufficiently strong statement does indeed follow from our Theorem 3.1 on the reduction of CM points at inert primes.

This theorem holds without the Heegner Hypothesis on N and K. We say that x = [E_{1} → E_{2}] ∈ X_{0}(N)(ℂ) is a CM point if E_{1} (and hence also E_{2}) has complex multiplication by K, i.e., if End_{ℂ}^{0}(E_{1}) = End_{ℂ}(E_{1}) ⊗ ℚ ≃ K; the Heegner points are special cases of CM points, for which we furthermore require that End_{ℂ}(E_{1}) = End_{ℂ}(E_{2}). Let E_{/ℂ} be an elliptic curve with CM by K, and C ⊂ E(ℂ) a cyclic subgroup of order N. Define L_{p} to be the set of all cyclic subgroups a ⊂ E(ℂ) of order p^{n} for some n ≥ 0. Since p ∤ N, we can associate to each a ∈ L_{p} a CM point H(a) = [E/a → E/(a ⊕ C)] ∈ X_{0}(N)(K[∞]). Let ℓ ∤ pN be a rational prime which is inert in K, and v_{ℓ} a place of K[∞] above ℓ. By Class Field Theory, the residue field k of v_{ℓ} is then isomorphic to 𝔽_{ℓ^{2}}. Theorem 3.1 states that under a technical assumption on the finite set R ⊂ Gal(K[∞]/K), the following map is surjective:

RED : L_{p} → X_{0}^{ss}(N)(k)^{R}
a ↦ (red_{ℓ}(σ · H(a)))_{σ∈R}

where red_{ℓ} : X_{0}(N)(K[∞]) → X_{0}(N)(k) is the reduction map at v_{ℓ} and X_{0}^{ss}(N)(k) is the set of supersingular points in X_{0}(N)(k).

The technical assumption alluded to above is needed to avoid the behaviour that we saw with G_{1}. It is verified for any set of representatives of G_{0}/G_{1}, so that by the argument sketched above, Mazur’s conjecture easily follows from the surjectivity of RED (take ℓ ≫ 0). However, to get the most out of this surjectivity, we furthermore use a theorem of Ihara to obtain refined and generalized versions of Mazur’s conjecture – Theorem A and B of section 4. For instance, we find:

Theorem. Assume that the kernel of π_{∗} : J_{0}(N) = Pic^{0}(X_{0}(N)) → 𝔼 is connected. Then for all prime q ∤ # (ℤ/Nd_{K}ℤ)^{∗}, the 𝔽_{q}-vector span of

{Tr_{K[p^{∞}]/H_{∞}}(y_{p^{n}}) ⊗ 1 | n ≥ 0} ⊂ 𝔼(H_{∞}) ⊗ 𝔽_{q}

has infinite dimension.

To prove Theorem 3.1, we first rewrite the map RED in p-adic terms, by means of a well-known adelic description of X_{0}^{ss}(N)(k) (section 2). Care must be taken to keep track of the Galois action on CM points. Theorem 3.1 then reduces to the surjectivity of a map

P SL_{2}(ℚ_{p}) → ∏_{σ∈R} P SL_{2}(ℤ_{p}) \ P SL_{2}(ℚ_{p})/Γ_{σ}^{1}
x ↦ ([x], ..., [x])

where Γ_{σ}^{1} is a discrete and cocompact subgroup of P SL_{2}(ℚ_{p}) associated to σ ∈ R. This very same map also arises in the context of Gross points on definite Shimura curves [19], and we then follow Vatsal’s proof with only minor improvements: put G = P SL_{2}(ℚ_{p}), Γ^{1} = ∏_{σ∈R} Γ_{σ}^{1}, and let Δ be the diagonal in G^{R}. The asserted surjectivity follows from the topological statement that ΔΓ^{1} is dense in G^{R}. A theorem of M. Ratner [14] states that the closure of ΔΓ^{1} equals HΓ^{1}, for a closed subgroup H ⊃ Δ of G^{R}. A purely group-theoretical result (section 3.6) forces H to be equal to the full group G^{R}, as soon as the Γ_{σ}^{1}’s are non-commensurable. The technical assumption on R guarantees just this non-commensurability.

Acknowledgement: It is a pleasure to thank Norbert Schappacher for guiding me through the realms of complex multiplication, Bas Edixhoven and Jan Nekovář for their patience and constant support. I have also benefited from many valuable suggestions by Jean-Louis Colliot-Thélène, and Julien Haristoy carefully read the manuscript. Last but not least, I am indebted to Nike Vatsal for the key idea to use Ratner’s Theorem and for freely sharing his work with me during a visit to Strasbourg.

Notations: We write \overline{F} for an algebraic closure of a field F, and F^{ab} ⊂ \overline{F} for its maximal abelian subextension. We fix once and for all an algebraically closed field Ω ⊃ K (Ω = ℂ), and for F ⊂ Ω we take \overline{F} within Ω. For a commutative group M and a prime number p, we put M̂ = M ⊗ ℤ̂, M_{p} = M ⊗ ℤ_{p} and M̂^{(p)} = M ⊗ ℤ̂^{(p)}, where ℤ̂, ℤ_{p} and ℤ̂^{(p)} are the profinite, p-adic and “prime-to-p-adic” completions of ℤ.

The formalism of 𝔞-Transforms: If R is a ring (unitary, but not necessarily commutative) acting on a commutative group scheme G_{/S}, then for any left R-module M, we define a presheaf of abelian group G^{M} on the category of S-schemes by the rule:

T_{/S} ⟼ G^{M}(T) = Hom_{R} (M, G(T)) .

This construction is covariant and left exact in G, contravariant and left exact in M. If G_{/S} is an abelian scheme, and M is a finite type left R-module, then G_{/S}^{M} is a proper commutative group scheme. If M is furthermore projective (resp. locally free of rank r), then G_{/S}^{M} is an abelian scheme (resp. of relative dimension r × dim(G_{/S})). For a left R-ideal I ⊂ R, we put G[I] = G^{R/I} — References: [16], [3].

1 CM points

1.1 Normalization of K ≃ End0 ; \overline{F} (E)

We say that an elliptic curve E over a field F has complex multiplication (CM) by K if End0 ; \overline{F} (E) is isomorphic to K. If F is a subfield of Ω, the action of End\overline{F} (E) on the one dimensional Ω-vector space Lie(E)(Ω) induces an embedding End0 ; \overline{F} (E) ↪ Ω which is onto K. We shall always identify K and End0 ; \overline{F} (E) in this way. End\overline{F} (E) is then an order O_{c} ⊂ K for a positive integer c, the conductor of E, and we also say that E has CM by O_{c}. Furthermore:

K ⊂ F ⟺ E has CM by K over F (i.e. End_{F} (E) = O_{c}).

With this normalization, any isogeny f : E_{1} → E_{2} between elliptic curves with CM by O_{c_{1}} and O_{c_{2}} respectively over a subfield F of Ω commutes with O_{c_{1}} ∩ O_{c_{2}}: we say that f is K-linear.

1.2 Families of isogenous points in X_{0}(N)

Let E_{/F} be an elliptic curve, with F ⊂ Ω. Let T̂(E) = \underleftarrow{lim} E[n](\overline{F}) = ∏_{q} T_{q}(E)

be its Tate module, and V̂ (E) = T̂(E) ⊗ ℚ, so that V̂ (E) is the restricted product of V_{q}(E) = T_{q}(E) ⊗ ℚ with respect to T_{q}(E), as q varies over the set of all rational primes. The inductive limit of

n^{−1}T̂(E)/T̂(E) ⟶^{∼} E(\overline{F})[n]

is a Galois equivariant isomorphism

V̂ (E)/T̂(E) ⟶^{∼} E(\overline{F})_{tors}.

In particular, if L is the set of ℤ̂-submodules of V̂ (E) that contains T̂(E) with a finite index, we obtain a Galois equivariant bijection between L and the set of finite subgroups of E(\overline{F}). For a ∈ L, we denote by X_{a} the corresponding subgroup of E(\overline{F})_{tors}, and put d(a) = #X_{a}. Using the special element e = T̂(E) of L, we shall view (L, e) as a pointed indexing set, and we thus write T̂(a) = a to refer to the submodule of V̂ (E) indexed by a ∈ L; it admits a decomposition T̂(a) = ∏_{q} T_{q}(a).

To each a ∈ L, we can associate the \overline{F} -isogeny g_{a} : E → E/X_{a}. We have identifications:

V̂ (E)/T̂(E) ⟶^{∼}  E(\overline{F})_{tors}
↓  ↓ g_{a}
V̂ (E)/T̂(a) ⟶^{∼} (E/X_{a})(\overline{F})_{tors}

If T̂(a) is stable under Gal(\overline{F} /F′) for some algebraic extension F′ of F, then the same holds for X_{a}, which thus descends to a finite subgroup scheme of E_{/F′}. In this situation, we simply say that E/X_{a} and g_{a} are defined over F′.

Let furthermore C ⊂ E(\overline{F}) be a cyclic subgroup of order N ≥ 1, and define

L′ = {a ∈ L | gcd(d(a), N) = 1}.

Then for each a ∈ L′, E/X_{a} → E/(X_{a} ⊕ C) is a cyclic N-isogeny, hence defines a point

H(a) = [E/X_{a} → E/(X_{a} ⊕ C)] ∈ X_{0}(N)(\overline{F}).

We refer to the map H : L′ → X_{0}(N)(\overline{F}) as the family of points associated to E_{/F} and C.

1.3 Families of isogenous CM points

Assume now that E has complex multiplication by O_{c} over F ⊂ Ω, so that H is a family of CM points. For any integer d ≥ 1, we let F [d] ⊂ F^{ab} be the composite of F and K[d]. Since K[c] = K(j(E)) ⊂ F by [16], we thus have F [d] = F [lcm(c, d)].

For each a ∈ L, the conductor of a is the unique integer c(a) ≥ 1 such that:

∀q : {x ∈ K_{q} | xT_{q}(a) ⊂ T_{q}(a)} = (O_{c(a)})_{q}.

T_{q}(a) is then free of rank one over (O_{c(a)})_{q}, and E/X_{a} has complex multiplication by O_{c(a)}. If e′ ∈ L corresponds to C, and a belongs to L′, then E/(X_{a} ⊕C) has complex multiplication by O_{c′(a)}, where c′(a) = c(a) × (c(e′)/c(e)).

In particular, T_{q}(E) is free of rank one over (O_{c})_{q}. Since the action of Gal(\overline{F} /F) on T_{q}(E) is (O_{c})_{q}-linear, it factors through a morphism:

ρ_{q} : Gal(F^{ab}/F) → (O_{c})_{q}^{∗}.

This implies that E(\overline{F})_{tors} = E(F^{ab})_{tors}. If furthermore F is a number field, we use class field theory to view ρ_{q} as a morphism ρ_{q} : I_{F} → (O_{c})_{q}^{∗}, where I_{F} is the idele group of F. Then:

Proposition 1.1 If O_{c}^{∗} = {±1}, Gal(F^{ab}/F [c(a)]) fixes T̂(a) for all a ∈ L.

Proof: Put d = lcm(c, c(a)). We want: ∀σ ∈ Gal(F^{ab}/F [d]), σT̂(a) = T̂(a). There exists a continuous homomorphism ε : I_{F} → K^{∗} such that:

∀s ∈ I_{F} , ∀q : ρ_{q}(s) = ε(s)N_{F/K}(s_{q}^{−1}) ∈ (O_{c})_{q}^{∗},

where s_{q} is the q-component of s. This is Theorem 10 of [17] in case c = 1 (End_{F} (E) = O_{K}). The general case reduces to it, since there exists a finite subgroup D ⊂ E(\overline{F}), defined over F, and such that E/D has complex multiplication by O_{K} over F: simply take D = \{x ∈ E(\overline{F}) | cO_{K} · x = 0\}.

Let [F^{ab}/F, ⋆] : I_{F} → Gal(F^{ab}/F) be the Artin reciprocity map, and pick s ∈ I_{F} such that [F^{ab}/F, s] = σ ∈ Gal(F^{ab}/F [d]). Since

σ |_{K[d]}= 1 = [K[d]/K, N_{F/K}(s)] ,

N_{F/K}(s) belongs to the norm subgroup of I_{K} corresponding to the abelian extension K[d]/K, namely K^{∗}(Ô_{d}^{∗} × ℂ^{∗}). Writing N_{F/K}(s) = λ(x̂ × μ) with λ ∈ K^{∗}, x̂ ∈ Ô_{d}^{∗} ⊂ Ô_{c}^{∗} and μ ∈ ℂ^{∗}, we obtain:

∀q : ε(s)λ^{−1} = ρ_{q}(s)x̂_{q}

The l.h.s. belongs to K and the r.h.s to (O_{c})_{q}^{∗}, so that finally ε(s)λ^{−1} belongs to O_{c}^{∗} = {±1}. But then ρ_{q}(s) = ±x̂_{q}^{−1} ∈ (O_{d})_{q}^{∗}, hence ρ_{q}(s)T̂_{q}(a) = T̂_{q}(a) for all q since O_{d} ⊂ O_{c(a)}, so that indeed σT̂(a) = T̂(a). □

As a consequence, E → E/X_{a} is defined over F [c(a)], and for a ∈ L′, H(a) belongs to X_{0}(N)(F [lcm (c(a), c′(a))]). We shall refine this latter fact below.

1.4 The Galois action on CM points

1.4.1 Fields of definition

Proposition 1.2 Let x = [f : E_{1} → E_{2}] ∈ X_{0}(N)(Ω) be a CM point, with End(E_{1}) = O_{c_{1}} and End(E_{2}) = O_{c_{2}}. Put c = lcm(c_{1}, c_{2}). Let S be a finite set of rational primes subject to the condition: if d_{K} = −3 or −4 and c is a power of p, then p ∉ S. Then there exist elliptic curves E′_{1/K[c]} and E′_{2/K[c]} with good reduction above S, and a K[c]-isogeny f′ : E′_{1} → E′_{2}, whose base change to Ω fits in a commutative diagram

E′_{1} ⟶^{∼} E_{1}
f′ ↓  ↓ f
E′_{2} ⟶^{∼} E_{2}

In particular, x = [f′ : E′_{1} → E′_{2}] ∈ X_{0}(N)(K[c]).

Proof: Our assumption on S implies by a theorem of Serre-Tate [17, p. 507] that there exists an elliptic curve E_{/K[c]} with good reduction above S and complex multiplication by O_{c}. Since E_{/Ω} is isogenous to E_{1} and E_{2}, we can find two subgroups H_{1} and H_{2} of E(K[c]^{ab}) and a commutative diagram of Ω-isogenies:

E/H_{1} ⟶^{∼} E_{1}
f′ ↓  ↓ f
E/H_{2} ⟶^{∼} E_{2}

If d_{K} = −3, −4 and c = 1, we can take H_{1} = H_{2} = 0 since Pic(O_{K}) = {1}; then f′ belongs to End_{K[1]}(E). In all other cases, we take for f′ the projection induced by an inclusion H_{1} ⊂ H_{2}, and then apply Proposition 1.1. □

1.4.2 The Galois action

Let x = [f : E_{1} → E_{2}] ∈ X_{0}(N)(Ω) be a CM point, with End(E_{1}) = O_{c_{1}} and End(E_{2}) = O_{c_{2}}, so that x is rational over K[c] with c = lcm(c_{1}, c_{2}). Following Serre [16], we shall describe the action of Gal(K[c]/K) on x using the formalism of 𝔞-transforms (cf. Introduction).

Recall the isomorphism from class field theory:

(\frac{K[c]/K}{⋆} ) : Pic(O_{c}) ⟵^{∼} K̂^{∗}/K^{∗}Ô_{c}^{∗} ⟶^{∼} Gal(K[c]/K).

Proposition 1.3 Let σ = (\frac{K[c]/K}{Q} ) for some O_{c}-proper fractional ideal Q of

K. Then: σ · x = [f^{Q} : E_{1}^{Q} → E_{2}^{Q}].

Proof: A straightforward generalization of the case N = 1, proven in [16]. □ Remark: Due to our chosen identification of K with End_{Ω}^{0}(E_{1}) and End_{Ω}^{0}(E_{2}), f is linear with respect to the action of O_{c} = O_{c_{1}} ∩ O_{c_{2}} on E_{1} and E_{2}, so that f^{Q} is well-defined, and ker(f^{Q})(Ω) = Hom_{O_{c}/NO_{c}}(Q/NQ, ker(f)(Ω)). Since Q/NQ ≃ O_{c}/NO_{c}, f^{Q} is indeed a cyclic N-isogeny. Remark: Let d ≥ 1 be an integer such that c | d, hence K[c] ⊂ K[d] and O_{d} ⊂ O_{c}. Then we can also describe the action of σ ∈ Gal(K[d]/K) on x using transforms with respect to the action of O_{d} on the elliptic curves, and an O_{d}-proper fractional ideal Q ⊂ K such that σ = (\frac{K[d]/K}{Q} ).

2 The supersingular locus X_{0}^{ss}(N)

Let N be a positive integer, ℓ ∤ N a prime number and k a finite field of characteristic ℓ. Let E_{/k} be a supersingular elliptic curve such that End_{k}(E) = End\overline{k}(E) = R. Then R is an order in B = End_{k}^{0}(E), a quaternion algebra that ramifies exactly at ℓ and ∞ [18, V.3]. The aim of this section is to describe the supersingular locus X_{0}^{ss}(N)(\overline{k}) of X_{0}(N)(\overline{k}) with these data, using the formalism of 𝔞-transforms (cf. Introduction).

2.1 Step 1: Inclusions of left R-ideals

We refer to nonzero finite type left R-submodules of B as left R-ideals. The right order of such an ideal I is O_{r}(I) = {x ∈ B | Ix ⊂ I} ≃ End_{R}(I). For left R-ideals I ⊂ J, we write I ⊂_{N,N} J as a shorthand for: J/I ≈ (ℤ/Nℤ)^{2}. Let Cl(R, N) be the orbit set for the obvious right action of B^{∗} on the set of those inclusions. We will prove below the following:

Proposition 2.1 R is a maximal order in B, all supersingular points of X_{0}(N)(\overline{k}) are rational over k and there exists a well-defined bijection

Cl(R, N) → X_{0}^{ss}(N)(\overline{k})
[I ⊂_{N,N} J] ↦ [E^{J} → E^{I}]

We first assume the (well-known) fact that R is a maximal order. Then:

Lemma 2.2 1. For any left R-ideal I, E_{/k}^{I} is a supersingular elliptic curve.

2. For any left R-ideals I and J, E^{⋆} : Hom_{R}(I, J) → Hom_{k}(E^{J}, E^{I}) is an isomorphism, and Hom_{k}(E^{J}, E^{I}) = Hom\overline{k}(E^{J}, E^{I}).

3. Any supersingular elliptic curve E′_{/\overline{k}} is isomorphic to E_{/\overline{k}}^{I} for some I.

4. If 0 → N → M → P → 0 is an exact sequence of finite type left Rmodules, then 0 → E^{P} → E^{M} → E^{N} → 0 is fppf-exact.

5. If M is a finite left R-module, then E_{/k}^{M} is a finite commutative group scheme of rank \sqrt{#M}.

6. For any nonzero R-linear map f : I → J between left R-ideals,

E^{f} : E^{J} → E^{I} is a cyclic N-isogeny ⟺ f(I) ⊂_{N,N} J.

Proof: 1-3) Every left R-ideal I is locally principal, hence defines an elliptic curve E_{/k}^{I}, which is supersingular since E^{I}(\overline{k})[ℓ] = Hom_{R}(I, E(\overline{k})[ℓ]) = 0. Since the number of isomorphism classes of left R-ideals equals the number of supersingular j-invariant (see [5, p. 117] and [18, V.4]), 3) follows from 2). Since Hom_{k}(E^{J}, E^{I}) is a direct factor of Hom\overline{k}(E^{J}, E^{I}) (a ℤ-module of rank 4), the injectivity of E^{⋆}, which is easy, implies the last equality of 2). It remains to show that E^{⋆} is surjective. When I = J, this follows from the fact that O_{r}(I) is a maximal order. In the general case, we may thus replace (R, E) by (O_{r}(J), E^{J}) and use the identifications

Hom_{R}(I, J) ≃ Hom_{O_{r}(J)}(J^{−1}I, O_{r}(J)) and E^{I} ≃ (E^{J})^{J^{−1}I},

(where J^{−1} = {x ∈ B | JxJ ⊂ J}) to reduce to the case where J = R. It is then routine to check that the subset Hom_{k}(E, E^{I}) of E^{I}(E) = Hom_{R}(I, E(E)) equals Hom_{R}(I, End_{k}(E)) = Hom_{R}(I, R).

4) We want: E^{M} → E^{N} is faithfully flat. By a formal argument, we may assume that N = I is a left R-ideal within M = R. Then E^{R} = E → E^{I} is an isogeny (hence faithfully flat), since its kernel E^{R/I} is finite.

5) Considering a surjective map R^{n}/#MR^{n} → M, we see that the rank of E_{/k}^{M} divides a power of #M. The statement is then a formal consequence of the fact that both rank(E_{/k}^{M}) and \sqrt{#M} are multiplicative on exact sequences of finite left R-modules by 4), and agree with the reduced norm of α when M = R/Rα, α ∈ R, α ≠ 0.

6) Put M = J/f(I), so that ker(E^{f}) ≃ E^{M}. Since E^{M}[d] ≃ E^{M/dM} for any positive integer d, we obtain using 5): E^{J} → E^{I} is a cyclic N-isogeny ⇔ ∀d, rank(E^{M}[d]) = gcd(N, d) ⇔ ∀d, #M/dM = gcd(N, d)^{2}. The result easily follows. □

We may now prove Proposition 2.1: still assuming that R is a maximal order, part 2) and 6) of the Lemma imply that the map [I ⊂_{N,N} J] ∈ Cl(R, N) ↦ [E^{J} → E^{I}] ∈ X_{0}^{ss}(N)(k) is well-defined and injective, whereas part 2), 3) and 6) imply that it is onto X_{0}^{ss}(N)(\overline{k}). Finally, to prove that R is indeed maximal, we merely need to construct a supersingular elliptic curve E′_{/k} whose endomorphism ring R′ is a maximal order, since the lemma then implies that the endomorphism ring of any supersingular elliptic curve is isomorphic to the right order of a left R′-ideal, hence maximal. But for any maximal order R′ in B, the reduced connected component of E^{RR′} is easily seen to be such an elliptic curve.

We shall also need the following straigthforward corollary of Lemma 2.2:

Lemma 2.3 Let h : E → E′ be a k-isogeny and I = {α ∈ R | α · ker(h) = 0}. Then E[I] = ker(h), so that there exists a commutative diagram

\xymatrix{E \ar[r] \ar[dr]_{h} & E^{I} \ar[d]^{≃} \\  & E′}

2.2 Step 2: Adelisation

Since R is finite over ℤ, R̂ = ∏_{q} R_{q}, and B̂ = ∏′_{q} B_{q} is the restricted product of the B_{q}’s with respect to the R_{q}’s, as q varies over all prime numbers. For a left R-ideal I and a finite idele b̂ = (b_{q}) ∈ B̂^{∗}, we define the left R-ideal Ib̂ ⊂ B by the rule: (Ib̂)_{q} = I_{q}b_{q} for all q. Equivalently, Ib̂ = Îb̂ ∩ B in B̂. The right order of Ib̂ is O_{r}(Ib̂) = b̂^{−1}O_{r}(I)b̂. If I ⊂_{N,N} J, then Ib̂ ⊂_{N,N} Jb̂ , and we thus obtain a right action of B̂^{∗} (extending that of B^{∗}) on the set of those inclusions. The stabilizer of (I ⊂_{N,N} J) is easily seen to be { \atop O_{r}(J/I)}^{∗}, where O_{r}(J/I) = O_{r}(J) ∩ O_{r}(I) (an Eichler order). Furthermore:

Lemma 2.4 This action is transitive.

Proof: Let (I ⊂_{N,N} J) and (I′ ⊂_{N,N} J′) be inclusions of left R-ideals. The question is local, so we must show that for any q, there exists b_{q} ∈ B_{q}^{∗} such that (I′_{q} ⊂ J′_{q})b_{q} = (I_{q} ⊂ J_{q}). Note that if q^{r} exactly divides N, then I′_{q} ⊂_{q^{r} ,q^{r}} J′_{q} and I_{q} ⊂_{q^{r} ,q^{r}} J_{q}. If q = ℓ, let π ∈ R_{ℓ} be a uniformising element; since left R_{ℓ}-ideals can be written as R_{ℓ}π^{n} for some n, the existence of b_{ℓ} is clear. If q ≠ ℓ, then R_{q} ≈ M_{2}(ℤ_{q}) and the Morita correspondence reduces the problem to a similar and classical statement on lattices in ℤ_{q}^{2}. □ Thus, fixing an inclusion of left R-ideals (I_{0} ⊂_{N,N} J_{0}), we obtain a bijection:

{ \atop O_{r}(J_{0}/I_{0})}^{∗} \ B̂^{∗}/B^{∗} ⟶^{∼} Cl(R, N).

Since ℤ̂^{∗}ℚ^{∗} = ℚ̂^{∗} (ℤ is principal),

{ \atop O_{r}(J_{0}/I_{0})}^{∗} \ B̂^{∗}/B^{∗} = { \atop O_{r}(J_{0}/I_{0})}^{∗}ℚ̂^{∗} \ B̂^{∗}/B^{∗}.

2.3 Step 3: Strong approximation

Fix a prime number p ≠ ℓ and define R_{0} = O_{r}(J_{0}/I_{0}), an (Eichler) order in B. The strong approximation theorem [21, p. 81] implies that the embedding B_{p}^{∗} → B̂^{∗} induces a surjective map B_{p}^{∗} → R̂_{0}^{∗}ℚ̂^{∗} \ B̂^{∗}/B^{∗}. If Γ is the image of R_{0}[1/p]^{∗} in B_{p}^{∗}, an easy computation shows that:

R_{0,p}^{∗}ℚ_{p}^{∗} \ B_{p}^{∗}/Γ ⟶^{∼} R̂_{0}^{∗}ℚ̂^{∗} \ B̂^{∗}/B^{∗}.

We finally obtain a sequence of bijections:

\begin{matrix}R_{0,p}^{∗}ℚ_{p}^{∗} \ B_{p}^{∗}/Γ & → & R̂_{0}^{∗}ℚ̂^{∗} \ B̂^{∗}/B^{∗} & → & Cl(R, N) & → & X_{0}^{ss}(N)(k) \\ [b ∈ B_{p}^{∗}] & ↦ & [b ∈ B̂^{∗}] & ↦ & [I_{0}b ⊂ J_{0}b] & ↦ & [E^{J_{0}b} → E^{I_{0}b}] \end{matrix}

3 The reduction of CM points at inert primes

3.1 Notations and Result

Let E_{/Ω} be an elliptic curve with complex multiplication by K, together with a cyclic subgroup C ⊂ E(Ω) of order N. Let

H : L′ → X_{0}(N)(K[∞])

be the associated family of CM points, where K[∞] = ⋃_{c≥1} K[c] – see section 1 for the definition, and all related notations. We do not require any hypothesis on N relative to K in this section. Let p ∤ N be a prime number, and define

L_{p} = {a ∈ L | X_{a} ≈ ℤ/p^{n}ℤ for some n ≥ 0} ⊂ L′.

Let S be a finite set of rational primes ℓ ∤ Np which are inert in K; choose for each ℓ ∈ S a place v_{ℓ} of K[∞] above ℓ, and let k(ℓ) be its residue field, so that k(ℓ) ≈ 𝔽_{ℓ^{2}}. Since ℓ ∤ N is inert in K, the reduction map at v_{ℓ},

red_{ℓ} : X_{0}(N)(K[∞]) → X_{0}(N)(k(ℓ)),

maps any CM point (relative to K) to the supersingular locus X_{0}^{ss}(N)(k(ℓ)) of X_{0}(N)(k(ℓ)) [3, 3.3.4]. Finally, let R be a finite subset of Gal(K[∞]/K).

Denote [K[∞]/K, ⋆] : K̂^{∗} ↠ Gal(K[∞]/K) the Artin reciprocity map. This section will be devoted to the proof of the following theorem.

Theorem 3.1 Assume that ∀(σ ≠ σ′) ∈ R^{2}, σ^{−1}σ′ ∉ [K[∞]/K, K̂^{(p)∗}]. Then

RED : L_{p} → ∏_{ℓ∈S} X_{0}^{ss}(N)(k(ℓ))^{R}
a ↦ (red_{ℓ}(σ · H(a)))_{σ∈R,ℓ∈S}

is surjective.

We refer the reader to the Introduction for an overview of the proof. With R = {1}, the theorem implies:

Corollary 3.2 Let I be the set of all rational primes ℓ ∤ pN inert in K, choose for each ℓ a place v_{ℓ} of K[∞] above ℓ and let k(ℓ) be the residue field. Then the image of the map

a ∈ L_{p} ↦ (red_{ℓ}(H (a) ))_{ℓ∈I} ∈ ∏_{ℓ∈I} X_{0}^{ss}(N)(k(ℓ))

is dense (with respect to the product of the discrete topologies).

3.2 Preliminary normalizations

3.2.1 Fields of definition

In view of Proposition 1.2, we first may and do assume that E is defined over a finite extension F ⊂ K[∞] of K, that E_{/F} has good reduction at all places above S, and is furthermore F -isogenous to an elliptic curve with complex multiplication by O_{c′}, with c′ prime to S and O_{c′}^{∗} = {±1} (the latter is an empty condition if d_{K} ≠ −3, −4).

If e, e′ ∈ L correspond respectively to the subgroups 0 and C of E(\overline{F}), then for each a ∈ L′, E/X_{a} has CM by O_{c(a)} and E/(X_{a} ⊕ C) has CM by O_{c′(a)}, where c′(a) = c(a) × c(e)/c(e′). Propositions 1.1 and 1.2 imply:

• g_{a} : E → E/X_{a} is defined over F [c(a)],

• h_{a} : E/X_{a} → E/(X_{a} ⊕ C) is defined over F [lcm(c(a), c′(a))],

• H(a) = [E/X_{a} → E/(X_{a} ⊕ C)] belongs to X_{0}(N)(K[lcm(c(a), c′(a))]).

3.2.2 Good reduction

Let v be a place of K[∞] with residue field k(ℓ) of characteristic ℓ inert in K. Let F_{1} ⊂ F_{2} be two finite subextensions of K[∞]/K, with valuation rings O_{v_{1}} ⊂ O_{v_{2}} at v. Let A_{/F_{1}} be an abelian variety with good reduction at v, so that its Néron model A_{/O_{v_{1}}} is an abelian scheme. The base change of this Néron model to O_{v_{2}} is also an abelian scheme, hence the Néron model of its generic fiber A_{/F_{2}}, which is the base change of A_{/F_{1}} to F_{2}. Since the residue fields of O_{v_{1}} and O_{v_{2}} are both equal to k(ℓ) (ℓ being inert in K), we can simply identify the special fibers of the Néron models of A_{/F_{1}} and A_{/F_{2}}. We will refer to this special fiber Ã_{/k(ℓ)} as the “reduction of A at v”, disregarding the subextension F of K[∞]/K where we need to consider A.

Since E_{/F} has good reduction at all places above ℓ ∈ S, this applies to the elliptic curves that we shall consider below, these being isogenous to E over finite subextensions of K[p^{∞}]/F. We can thus refer to their reduction at v_{ℓ}.

3.2.3 Normalizing data

The computations below involve a few normalizations with respect to our base point e, ultimately reducing to the choice of an isomorphism:

ξ : T_{p}(E) ⟶^{∼} ℤ_{p}^{2}.

It extends to an isomorphism ξ : V_{p}(E) ⟶^{∼} ℚ_{p}^{2}.

Let T_{p} = GL_{2}(ℤ_{p})ℚ_{p}^{∗} \ GL_{2}(ℚ_{p}) be the Bruhat-Tits tree of P GL_{2}(ℚ_{p}). For each a ∈ L_{p}, we choose τ_{a} ∈ GL_{2}(ℚ_{p}) such that τ_{a} · ℤ_{p}^{2} = ξ(T_{p}(a)) ⊂ ℚ_{p}^{2}. This yields a bijection:

\begin{aligned}ϕ_{1} : & L_{p} & ⟶^{∼} & T_{p} \\ & a & ⟼ & [τ_{a}^{−1}] \end{aligned}

For each ℓ ∈ S, let Ẽ_{/k(ℓ)} be the reduction of E at v_{ℓ}, R(ℓ) = End_{k(ℓ)}(Ẽ) and B(ℓ) = End_{k(ℓ)}^{0}(Ẽ). Our assumptions on E, F and S imply that R(ℓ) = End\overline{k(ℓ)}(Ẽ), so that R(ℓ) is a maximal order in B(ℓ), a quaternion algebra that ramifies precisely at ℓ and ∞. According to Lemmas 2.2 and 2.3, the reduction of h : E → E/C at v_{ℓ} is a (separable) cyclic N-isogeny with kernel C̃, which identifies with the isogeny Ẽ → Ẽ^{I(ℓ)} induced by I(ℓ) ⊂_{N,N} R(ℓ), where

I(ℓ) = {x ∈ R(ℓ) | x · C̃ = 0} .

Using Ẽ_{/k(ℓ)} and this inclusion of left R(ℓ)-ideals to normalize the description of X_{0}^{ss}(N)(k(ℓ)) in section 2, we thus obtain:

R(ℓ)_{p}^{∗}ℚ_{p}^{∗} \ B(ℓ)_{p}^{∗}/Γ(ℓ) ⟶^{∼} X_{0}^{ss}(N)(k(ℓ))
[b ∈ B(ℓ)_{p}^{∗}] ⟼ [Ẽ^{R(ℓ)·b} → Ẽ^{I(ℓ)·b}]

where Γ(ℓ) = R′(ℓ)[1/p]^{∗} ⊂ B(ℓ)_{p}^{∗}, with R′(ℓ) = R(ℓ) ∩ O_{r}(I(ℓ)) (observe that R′(ℓ)_{p} = R(ℓ)_{p} since p does not divide N).

The reduction at v_{ℓ} however yields an isomorphism x ∈ T_{p}(E) ↦ x̃ ∈ T_{p}(Ẽ), which together with ξ gives an isomorphism ξ̃: T_{p}(Ẽ) → ℤ_{p}^{2}. Using the action of R(ℓ)_{p} on T_{p}(Ẽ), we thus obtain an isomorphism θ_{ℓ} : B(ℓ)_{p} → M_{2}(ℚ_{p}) mapping R(ℓ)_{p} onto M_{2}(ℤ_{p}), and characterized by θ_{ℓ}(α) · ξ(x) = ξ̃(α · x̃) for all α ∈ R(ℓ), x ∈ T_{p}(E). If i_{ℓ} : K → B(ℓ) is the natural embedding given by the reduction of endomorphisms, then the composite map θ_{ℓ} ∘ i_{ℓ} : K_{p} → M_{2}(ℚ_{p}) does not depend on ℓ ∈ S.

Using θ_{ℓ}, we can now rewrite our description of X_{0}^{ss}(N)(k(ℓ)) as a bijection:

T_{p}/θ_{ℓ}(Γ(ℓ)) ⟶^{∼} X_{0}^{ss}(N)(k(ℓ))
[θ_{ℓ}(b)] ⟼ [Ẽ^{R(ℓ)·b} → Ẽ^{I(ℓ)·b}]

(for b ∈ B(ℓ)_{p}^{∗}). Let δ_{ℓ} be the inverse map.

Finally, we choose for each σ ∈ R a finite idele λ̂_{σ} ∈ K̂^{∗} such that

[K[∞]/K, λ̂_{σ}] = σ ∈ Gal(K[∞]/K).

3.3 Computation of red_{ℓ}(σ · H(a))

We first compute the (ℓ, σ)-component of RED (ℓ ∈ S, σ ∈ R), and drop the fixed ℓ from all the above notations: R = R(ℓ), B = B(ℓ), θ = θ_{ℓ} and so on.

3.3.1 Step 1

Pick a ∈ L′ and start with

H(a) = [h_{a} : E/X_{a} → E/X_{a} ⊕ C] ∈ X_{0}(N)(K[ lcm(c(a), c′(a))]).

Let d = lcm(c(a), c′(a), c(e)) and Q = O_{d} · λ̂_{σ} ⊂ K. Then Q is a proper O_{d}-ideal and σ |_{K[d]}= (\frac{K[d]/K}{Q} ) ∈ Gal(K[d]/K), hence by proposition 1.3:

σ · H(a) = [h_{a}^{Q} : (E/X_{a})^{Q} → (E/X_{a} ⊕ C)^{Q}] ∈ X_{0}(N)(K[d]),

where we use the action of O_{d} ⊂ O_{c(a)}, O_{c′(a)} on the elliptic curves to define their transforms. Reducing at v_{ℓ} we find

red_{ℓ}(σ · H(a)) = [((E/X_{a})^{Q})^{∼} → ((E/X_{a} ⊕ C)^{Q})^{∼}]
= [((E/X_{a})^{∼})^{Q} → ((E/X_{a} ⊕ C)^{∼})^{Q}] .

Let J_{a} = {x ∈ R | x · (X_{a})^{∼} = 0} and I_{a} = {x ∈ R | x · (X_{a} ⊕ C)^{∼} = 0}. Thus J_{e} = R and I_{e} = I. According to the lemmas 2.2 and 2.3, I_{a} ⊂_{N,N} J_{a}, [R : J_{a}] = [I : I_{a}] = d(a)^{2} and there exists a commutative diagram

Ẽ^{J_{a}}  ⟶^{∼}  (E/X_{a})^{∼}
↓  ↓
Ẽ^{I_{a}}  ⟶^{∼} (E/X_{a} ⊕ C)^{∼} .

By transport de structure, our point can thus be written:

red_{ℓ}(σ · H(a)) = [\binom{}{E^{J_{a}}}^{Q} → \binom{}{E^{I_{a}}}^{Q}] ∈ X_{0}^{ss}(N)(k(ℓ)).

However, we have to know how O_{d} acts on Ẽ^{J_{a}} and Ẽ^{I_{a}}. Unwinding the definitions, we find that for any k(ℓ)-scheme T, in the formula

\binom{}{E^{J_{a}}}^{Q} (T) = Hom_{O_{d}} (Q, Hom_{R} (J_{a}, Ẽ(T))) ,

x ∈ O_{d} acts on Hom_{R} (J_{a}, Ẽ(T)) by right multiplication by i(x) on J_{a} (and i(O_{d}) ⊂ O_{r}(J_{a})): this follows from the fact that the isogenies that we are reducing are K-linear. Since Q is O_{d}-projective, we obtain:

\binom{}{E^{J_{a}}}^{Q} (T) = Hom_{R} (J_{a} ⊗_{i(O_{d})} i(Q), Ẽ(T))
= Hom_{R} (J_{a}i(Q), Ẽ(T))
= Ẽ^{J_{a}i(Q)}(T).

But J_{a}i(Q) = J_{a}i(O_{d}λ̂_{σ}) = J_{a} · i(λ̂_{σ}), so that

red_{ℓ}(σ · H(a)) = [Ẽ^{J_{a}·i(λ̂_{σ})} → Ẽ^{I_{a}·i(λ̂_{σ})}] ∈ X_{0}^{ss}(N)(k(ℓ)).

where we now use the action of R on Ẽ to construct the indicated isogeny.

3.3.2 Step 2

Assume from now on that a belongs to L_{p}, let d(a) = p^{n} and define x_{a} ∈ B_{p}^{∗} by θ(x_{a}) = τ_{a}^{−1}. Then, viewing x_{a} as an element of B̂^{∗} we have:

Lemma 3.3 J_{a} = R · x_{a} and I_{a} = I · x_{a}.

Proof: Both sides of the first equality do not differ from R outside p, so we just need to check that they are also equal at p. Reducing the commutative diagram

T_{p}(a)/T_{p}(E) ⟶^{∼} X_{a}
∩  ∩
V_{p}(E)/T_{p}(E) ⟶^{∼} E(\overline{F})_{p−tors}

we find that

(J_{a})_{p} = {x ∈ B_{p} | x · (T_{p}(a))^{∼} ⊂ T_{p}(Ẽ)}
= {x ∈ B_{p} | ξ̃(x · (T_{p}(a))^{∼}) ⊂ ξ̃(T_{p}(Ẽ))}
= {x ∈ B_{p} | θ(x) · τ_{a} · ℤ_{p}^{2} ⊂ ℤ_{p}^{2}}
= {x ∈ B_{p} | θ(x) ∈ M_{2}(ℤ_{p})τ_{a}^{−1}}

since ξ̃((T_{p}(a))^{∼}) = τ_{a} · ℤ_{p}^{2} and ξ̃(x · t) = θ(x) · ξ̃(t) for x ∈ B_{p} and t ∈ V_{p}(Ẽ). But M_{2}(ℤ_{p}) = θ(R_{p}), so that (J_{a})_{p} = R_{p}x_{a} as was to be shown.

The second equality may be proven similarly, and follows anyway from the first one since p is prime to N. □

Our point thus becomes:

red_{ℓ}(σ · H(a)) = [Ẽ^{R·(x_{a}i(λ̂_{σ}))} → Ẽ^{I·(x_{a}i(λ̂_{σ}))}] ∈ X_{0}^{ss}(N)(k(ℓ)).

3.3.3 Step 3

The strong approximation theorem (section 2.3) implies that there exists an element b ∈ B^{∗} such that for all prime q ≠ p:

((I · i(λ̂_{σ}) ⊂_{N,N} R · i(λ̂_{σ})) × b)_{q} = (I ⊂_{N,N} R)_{q} .

Since x_{a} ∈ B_{p}^{∗}, we then also have:

((I · (x_{a}i(λ̂_{σ})) ⊂_{N,N} R · (x_{a}i(λ̂_{σ}))) × b)_{q} = (I ⊂_{N,N} R)_{q} .

The inclusion (I · (x_{a}i(λ̂_{σ})) ⊂ R · (x_{a}i(λ̂_{σ}))) × b can therefore be computed

using only the p-component of (x_{a}i(λ̂_{σ})) × b, and our point thus becomes:

red_{ℓ}(σ · H(a)) = [Ẽ^{R·(x_{a}i(λ̂_{σ,p})b)} → Ẽ^{I·(x_{a}i(λ̂_{σ,p})b)}] ∈ X_{0}^{ss}(N)(k(ℓ)),

where we now consider b as an element of B_{p}^{∗} ⊂ B̂^{∗}. In other words:

δ_{ℓ}(red_{ℓ}(σ · H(a))) = [τ_{a}^{−1}θ(i(λ̂_{σ,p})b)] ∈ T_{p}/θ(Γ).

3.4 Enters topology

We have just proven that ∀(ℓ, σ) ∈ S × R, there exists b_{ℓ,σ} ∈ B(ℓ)^{∗} such that:

∀a ∈ L_{p} : δ_{ℓ}(red_{ℓ}(σ · H(a))) = [τ_{a}^{−1}θ_{ℓ}(i_{ℓ}(λ̂_{σ,p})b_{ℓ,σ})] ∈ T_{p}/θ_{ℓ}(Γ(ℓ)).

Multiplication on the right by z_{ℓ,σ} = θ_{ℓ}(i_{ℓ}(λ̂_{σ,p})b_{ℓ,σ})^{−1} ∈ GL_{2}(ℚ_{p}) yields a bijection T_{p}/θ_{ℓ}(Γ(ℓ)) ⟶^{∼} T_{p}/Γ_{ℓ,σ}, where Γ_{ℓ,σ} = z_{ℓ,σ}^{−1}θ_{ℓ}(Γ(ℓ))z_{ℓ,σ}. Composing it with δ_{ℓ}, we obtain a bijection:

δ_{ℓ,σ} : X_{0}^{ss}(N)(k(ℓ)) ⟶^{∼} T_{p}/Γ_{ℓ,σ},

and δ_{ℓ,σ}(red_{ℓ}(σ · H(a))) = [τ_{a}^{−1}] is the class of ϕ_{1}(a) ∈ T_{p}.

We have thus constructed a commutative diagram:

RED :  L_{p} → ∏_{ℓ∈S} (X_{0}^{ss}(N)(k(ℓ)))^{R}
ϕ_{1} ↓ ≃  ≃ ↓ ϕ_{2}
DIAG :  T_{p} →  ∏_{ℓ∈S,σ∈R} T_{p}/Γ_{ℓ,σ}

where DIAG is the “diagonal map” and ϕ_{2} = (δ_{ℓ,σ})_{ℓ∈S,σ∈R}. Theorem 3.1 is therefore equivalent to the surjectivity of:

\begin{matrix}P GL_{2}(ℚ_{p}) & → & ∏_{ℓ,σ} P GL_{2}(ℤ_{p}) \ P GL_{2}(ℚ_{p})/Γ_{ℓ,σ} \\ v & ↦ & ([v], ⋯, [v]) \end{matrix}

In fact, we can work with P SL_{2}(ℚ_{p}) instead of P GL_{2}(ℚ_{p}). Indeed:

Proposition 3.4 P SL_{2}(ℚ_{p}) ↪ P GL_{2}(ℚ_{p}) induces a bijection:

P SL_{2}(ℤ_{p}) \ P SL_{2}(ℚ_{p})/Γ_{ℓ,σ}^{1} ⟶^{∼} P GL_{2}(ℤ_{p}) \ P GL_{2}(ℚ_{p})/Γ_{ℓ,σ},

where Γ_{ℓ,σ}^{1} is the intersection in P GL_{2}(ℚ_{p}) of P SL_{2}(ℚ_{p}) with the image of Γ_{ℓ,σ} ⊂ GL_{2}(ℚ_{p}).

This is a straightforward corollary of the following:

Lemma 3.5 ±p^{ℤ} ⊂ Γ_{ℓ,σ} and det(Γ_{ℓ,σ}) = p^{ℤ} for all ℓ ∈ S, σ ∈ R.

Proof: Since Γ_{ℓ,σ} = z_{ℓ,σ}^{−1}θ_{ℓ}(Γ(ℓ))z_{ℓ,σ}, we must show that ±p^{ℤ} ⊂ Γ(ℓ) and that the reduced norm of Γ(ℓ) equals p^{ℤ}. Since Γ(ℓ) = R′(ℓ)[1/p]^{∗}, the first statement is obvious, and the second follows from [21, p. 90]. □

So let G = P SL_{2}(ℚ_{p}), U = P SL_{2}(ℤ_{p})), call Δ : G → G^{S×R} the diagonal and put Γ^{1} = ∏_{ℓ,σ} Γ_{ℓ,σ}^{1} ⊂ G^{S×R}. The Γ_{ℓ,σ}^{1}’s are discrete and cocompact subgroups of G [21, p. 104], and U is an open compact subgroup. Theorem 3.1 is then equivalent to the statement that the “diagonal” map

G → U^{S×R} \ G^{S×R}/Γ^{1}

is surjective. Since U^{S×R} is an open subgroup of G^{S×R}, this follows from:

Proposition 3.6 Δ(G)Γ^{1} is dense in G^{S×R}.

We prove this proposition in the next three paragraphs, essentially following Vatsal ([19] and [20]).

3.5 Commensurability

We say that two subgroups H_{1} and H_{2} of G are commensurable, and write H_{1} ∼ H_{2}, if H_{12} = H_{1} ∩ H_{2} has finite index in H_{1} and H_{2}. For a subgroup H of G, the commensurator of H is Com(H, G) = {x ∈ G | x^{−1}Hx ∼ H}. If H_{1} ∼ H_{2}, then Com(H_{1}, G) = Com(H_{2}, G). Our assumption in Theorem 3.1 is precisely meant for the following:

Proposition 3.7 If (ℓ, σ) ≠ (ℓ′, σ′) ∈ (S × R)^{2}, then Γ_{ℓ,σ}^{1} and Γ_{ℓ′,σ′}^{1} are not commensurable in G.

Proof: First recall that Γ_{ℓ,σ}^{1} is the intersection in P GL_{2}(ℚ_{p}) of P SL_{2}(ℚ_{p}) with the image of Γ_{ℓ,σ} ⊂ GL_{2}(ℚ_{p}). For this step, we have:

Lemma 3.8 Let Γ_{1} and Γ_{2} be subgroups of GL_{2}(ℚ_{p}) such that det(Γ_{i}) = p^{ℤ} and ±p^{ℤ} ⊂ Γ_{i} for i = 1, 2. Let Γ_{i}^{1} be the intersection in P GL_{2}(ℚ_{p}) of P SL_{2}(ℚ_{p}) with the image of Γ_{i}, i = 1, 2. Then Γ_{1}^{1} ∼ Γ_{2}^{1} ⇔ Γ_{1} ∼ Γ_{2}.

By lemma 3.5, the proposition is therefore equivalent to a similar statement about the Γ_{ℓ,σ}’s. The commensurator of Γ(ℓ) is known; we just quote the result from [21, p. 106] using our notations.

Proposition. Com (Γ(ℓ), B(ℓ)_{p}^{∗}) = ℚ_{p}^{∗}B(ℓ)^{∗}.

The commensurator of Γ_{ℓ,σ} = z_{ℓ,σ}^{−1}θ_{ℓ}(Γ(ℓ))z_{ℓ,σ} thus equals z_{ℓ,σ}^{−1}θ_{ℓ} (ℚ_{p}^{∗}B(ℓ)^{∗}) z_{ℓ,σ}.

First consider the case of a couple ((ℓ, σ), (ℓ′, σ′)) ∈ (S × R)^{2} with ℓ ≠ ℓ′. Then Γ_{ℓ,σ} and Γ_{ℓ′,σ′} are not commensurable since they simply do not have the same commensurator:

Proposition 3.9 Let B and B′ be two non-isomorphic quaternion algebras splitting at p, and choose identifications of B_{p} and B′_{p} with M_{2}(ℚ_{p}). Then:

ℚ_{p}^{∗}B^{∗} ≠ ℚ_{p}^{∗}B^{′∗} in GL_{2}(ℚ_{p}).

Proof: Suppose that ℚ_{p}^{∗}B^{∗} = ℚ_{p}^{∗}B^{′∗}, hence B ⊂ ℚ_{p}B′. For b ∈ B, write b = λb′ with λ ∈ ℚ_{p} and b′ ∈ B′, so that tr(b) = λtr(b′). If tr(b) ≠ 0, then tr(b′) ≠ 0 hence λ = tr(b)/tr(b′) ∈ ℚ, so that b ∈ B′. If tr(b) = 0, then tr(b−1) ≠ 0, hence b − 1 ∈ B′ and b ∈ B′. Thus B ⊂ B′; by symmetry B = B′, a contradiction. □

Consider then the case ℓ = ℓ′, and suppose that Γ_{ℓ,σ} ∼ Γ_{ℓ,σ′}, i.e. that

z_{ℓ,σ}z_{ℓ,σ′}^{−1} ∈ Com(θ_{ℓ}(Γ(ℓ)), GL_{2}(ℚ_{p})) = ℚ_{p}^{∗}θ_{ℓ}(B(ℓ)^{∗}).

Since z_{ℓ,σ} = θ_{ℓ}(i_{ℓ}(λ̂_{σ,p})b_{ℓ,σ})^{−1} with b_{ℓ,σ} ∈ B(ℓ)^{∗}, it follows that

i_{ℓ}(λ̂_{σ,p}^{−1}λ̂_{σ′,p}) ∈ ℚ_{p}^{∗}B(ℓ)^{∗} ⊂ B(ℓ)_{p}^{∗}.

Write i_{ℓ}(λ̂_{σ,p}^{−1}λ̂_{σ′,p}) = xy, with x ∈ ℚ_{p}^{∗} and y ∈ B(ℓ)^{∗}. Then y commutes with i_{ℓ}(K), a maximal commutative subring of B(ℓ), hence y belongs to i_{ℓ}(K) and

λ̂_{σ,p}^{−1}λ̂_{σ′,p} ∈ ℚ_{p}^{∗}K^{∗} ⊂ K_{p}^{∗}.

Since [K[∞]/K, ℚ_{p}^{∗}K^{∗}] = 1, it follows that σ^{−1}σ′ ∈ [K[∞]/K, K̂^{(p)∗}], so that σ = σ′ in view of our assumption in theorem 3.1. □

3.6 A lemma on simple non-commutative groups

Fix a group G, a finite set S, and put G^{S} = ∏_{s∈S} G. For s ∈ S, let p_{s} : G^{S} → G be the projection, and for S′ ⊂ S, identify G^{S′} to the corresponding subgroup of G^{S}: G^{S′} = {x ∈ G^{S} | ∀s ∈ S \ S′, p_{s}(x) = 1}. Denote Δ^{S′} : G → G^{S′} ⊂ G^{S} the diagonal of G^{S′}.

We say that a subgroup H of G^{S} is a product of diagonals if there exist a finite set I and disjoint subsets (S_{i})_{i∈I} of S such that

H = ∏_{i∈I} Δ^{S_{i}}(G) ⊂ G^{S}.

These subgroups are normalized by Δ^{S}(G). Conversely, we have the following result (a mild generalization of [20, Lemma 5.10]):

Proposition 3.10 If G is a simple non-commutative group, then any subgroup H of G^{S} which is normalized by Δ^{S}(G) is a product of diagonals.

Proof: 1) We use induction on n = #S. The starting case n = 1 is trivial since G is simple. We may thus assume that n > 1, and that the result is true for any set S′ of order n′ < n. If S′ ⊊ S, Δ^{S}(G) normalizes H and G^{S′}, hence also H^{S′} = H ∩G^{S′}. Then Δ^{S′}(G) also normalizes H^{S′}, which is therefore a product of diagonals in G^{S′} (and in G^{S}), by induction. Note also that ∀s ∈ S, p_{s}(H^{S′}) being normalized by p_{s}(Δ^{S}(G)) = G, is either {1} or G since G is simple.

Pick s_{0} ∈ S and put S_{0} = S \ {s_{0}}, so that H^{S_{0}} is a product of diagonals. If p_{s_{0}}(H) = {1}, then H = H^{S_{0}} and we are done. So we can assume that p_{s_{0}}(H) = G and pick a minimal subset S_{1} of S such that p_{s_{0}}(H^{S_{1}}) = G. Then s_{0} ∈ S_{1} and for any s ∈ S_{1}, p_{s_{0}}(H^{S_{1}\{s}}) = 1.

2) If S_{1} ≠ S, then H^{S_{1}} is a product of diagonals; the definition of S_{1} then implies: H^{S_{1}} = Δ^{S_{1}}(G). Suppose that S_{1} intersects the support S_{2} of a diagonal Δ^{S_{2}}(G) ⊂ H^{S_{0}}, and put S_{3} = (S_{1} ∪ S_{2}) \ (S_{1} ∩ S_{2}). Since S_{1} ∩ S_{2} ≠ ∅, S_{3} ⊊ S and H^{S_{3}} is a product of diagonals. If g ≠ 1 ∈ G, then z = Δ^{S_{2}}(g)Δ^{S_{1}}(g^{−1}) is a nontrivial element of H^{S_{3}}, so that H^{S_{3}} contains a nontrivial diagonal Δ^{S_{4}}(G). Since H^{S_{1}} = Δ^{S_{1}}(G) and H^{S_{2}} = Δ^{S_{2}}(G), S_{4} intersects both S_{1} and S_{2}. If s_{1} ∈ S_{1} ∩S_{4} and s_{2} ∈ S_{2} ∩S_{4}, then any element in H^{S_{3}} has the same component at s_{1} and s_{2}. In particular, p_{s_{1}}(z) = g^{−1} = p_{s_{2}}(z) = g, hence g^{2} = 1 for all g ∈ G. This forces G to be commutative, a contradiction. Thus S_{1} does not intersect the support of the diagonals of H^{S_{0}}; it follows easily that H is indeed a product of diagonals, namely those of H^{S_{0}}, and Δ^{S_{1}}(G).

3) If now S_{1} = S, p_{s_{0}}(H^{S\{s}}) = 1 for all s ∈ S, whereas p_{s_{0}}(H) = G. In particular, p_{s}(H) = G for all s ∈ S, since otherwise H = H^{S\{s}}, a contradiction. Moreover, H^{S_{0}} = 1: if H^{S_{0}} contains a nontrivial diagonal Δ^{S_{2}}(G), pick s ∈ S_{2}, and also x ∈ H such that p_{s_{0}}(x) ≠ 1; then z = xΔ^{S_{2}}(p_{s}(x)^{−1}) belongs to H^{S\{s}} and p_{s_{0}}(z) = p_{s_{0}}(x) ≠ 1, a contradiction. It follows that p_{s_{0}} : H → G is a bijection. Let q : G → H be the inverse map, and put θ_{s} = p_{s} ∘ q for s ∈ S, so that θ_{s} is a surjective homomorphism, and θ_{s_{0}} is the identity.

For all x, y ∈ G, Δ^{S}(y)^{−1}q(x)Δ^{S}(y) belongs to H, hence equals q(z) for some z ∈ G. Looking first at the s_{0}-component, we find that z = y^{−1}xy; looking then at the s-component, we find θ_{s}(y^{−1}xy) = y^{−1}θ_{s}(x)y. Since also θ_{s}(y^{−1}xy) = θ_{s}(y)^{−1}θ_{s}(x)θ_{s}(y), we obtain: θ_{s}(y)y^{−1}θ_{s}(x) = θ_{s}(x)θ_{s}(y)y^{−1}, so that θ_{s}(y)y^{−1} ∈ G commutes with all elements of θ_{s}(G) = G. Since G is simple and non-commutative, θ_{s}(y) = y, hence q(y) = Δ^{S}(y) and H = q(G) = Δ^{S}(G) is the full diagonal in G^{S}. □

Remark: This property characterizes the simple non-commutative groups.

3.7 An application of Ratner’s theorem

Proposition 3.11 [20, Lemma 5.10] Let G = P SL_{2}(ℚ_{p}) and X ≠ ∅ be a finite set. Let (Γ_{x})_{x∈X} be a collection of mutually non-commensurable discrete and cocompact subgroups of G, and put Γ = ∏_{x∈X} Γ_{x} ⊂ G^{X}. Then ΔΓ is dense in G^{X}, where Δ is the diagonal of G^{X}.

Proof: We use induction on n = #X. If n = 1, there is nothing to prove since Δ = G already. So let us assume that n > 1.

By Ratner’s Theorem on the closure of unipotent orbits in p-adic Lie groups [14, Theo. 2], there exists a closed subgroup H ⊂ G^{X} containing Δ, such that \overline{ΔΓ} = HΓ. Since G is simple and non-commutative, and Δ normalizes H, Proposition 3.10 says: H is a product of diagonals. Since Δ ⊂ H, this means that there exists a partition (X_{i})_{i∈I} of X such that, in the notations of 3.10, H = ∏_{i∈I} Δ^{X_{i}}(G).

Suppose that #I = 1, so that H = Δ and ΔΓ is already closed in G^{X}, hence a Baire space since G^{X} is locally compact. Now Γ is discrete in G^{X} (and therefore countable), so that Δ is open in ΔΓ and the natural continuous map

G/ ⋂_{x∈X} Γ_{x} → Δ/Δ ∩ Γ → ΔΓ/Γ

is a homeomorphism. It follows that G/ ⋂_{x∈X} Γ_{x} is compact since Γ is cocompact in G^{X}. But then for any x_{0} ∈ X, Γ_{x_{0}}/ ⋂_{x∈X} Γ_{x} should be both discrete and compact, hence finite: this contradicts our non-commensurability assumption, since n > 1.

Therefore #I ≠ 1 and for all i ∈ I, #X_{i} < n. According to the induction hypothesis, Δ^{X_{i}}(G)Γ^{X_{i}} is dense in the closed subgroup G^{X_{i}} of G^{X}, so that G^{X_{i}} ⊂ \overline{ΔΓ}, hence G^{X} = \overline{ΔΓ}. □

In view of Proposition 3.7, this proves Proposition 3.6 and therefore also Theorem 3.1.

4 Mazur’s Conjecture

Let 𝔸_{/ℚ} be a nonzero modular abelian variety, i.e., such that there exists a surjective ℚ-morphism α : J_{0}(N) → 𝔸. Define π : X_{0}(N) → 𝔸 to be the composite of α with the usual embedding X_{0}(N) → J_{0}(N) that sends ∞ to 0.

We assume that the Heegner Hypothesis holds for N and K (see the introduction), and choose an ideal N of O_{K} such that O_{K}/N ≃ ℤ/Nℤ. Let furthermore E_{/Ω} be an elliptic curve with complex multiplication by O_{K}, and consider the family of Heegner points associated to E and C = E[N] (see section 1):

H : a ∈ L′ ⟼ H(a) = [E/X_{a} → E/X_{a} ⊕ C] ∈ X_{0}(N)(K[∞]).

Let p ∤ N be a prime number and put G_{0} = Gal(K[p^{∞}]/K)_{tors}, so that G_{0} = Gal(K[p^{∞}]/H_{∞}). If R is a ring and χ : G_{0} → R^{∗} is a character, we write e_{χ} = ∑_{σ∈G_{0}} χ^{−1}(σ)σ ∈ R[G_{0}] for the corresponding “idempotent”.

The Heegner point x_{p^{n}} of the Introduction belongs to the sub-family H(L_{p}). Moreover, it is not hard to see — using for instance Proposition 1.3, that

H(L_{p}) ⊂ {σx_{p^{n}} | σ ∈ Gal(K[p^{∞}]/K), n ≥ 0} ⊂ X_{0}(N)(K[p^{∞}]).

Mazur’s conjecture thus follows (with χ = 1) from the stronger statement:

Theorem A. For any character χ : G_{0} → ℂ^{∗}, the ℂ-vector span H_{χ} of

{e_{χ} (π(H(a)) ⊗ 1) | a ∈ L_{p}} ⊂ 𝔸 (K[p^{∞}]) ⊗ ℂ

has infinite dimension.

We first remark that 𝔸 (K[p^{∞}])_{tors} is finite, since:

Lemma 4.1 𝔸 (K[∞])_{tors} is finite.

Proof: [11] Let v be a place of K[∞] above ℓ ∤ N inert in K. Since 𝔸_{/ℚ} has good reduction at ℓ (being a quotient of J_{0}(N)_{/ℚ}), reduction at v yields an injective map 𝔸 (K[∞])_{non−ℓ−tors} → 𝔸(𝔽_{ℓ^{2}}). Using a second place v′ above ℓ′ ≠ ℓ shows that 𝔸 (K[∞])_{tors} is indeed finite. □

Theorem A is therefore a consequence of:

Theorem B. If q is a prime number dividing neither φ(Nd_{K}) = #(ℤ/Nd_{K}ℤ)^{∗}, nor the number η of geometrically connected components of ker(α), then for any character χ : G_{0} → 𝔽_{q}^{∗}, the 𝔽_{q}-vector span H_{χ} of

{e_{χ} (π(H(a)) ⊗ 1) | a ∈ L_{p}} ⊂ 𝔸 (K[p^{∞}]) ⊗ 𝔽_{q}

has infinite dimension.

Indeed, starting with a character χ : G_{0} → ℂ^{∗}, let O_{χ} be the ring of integers of ℚ(χ(G_{0})). For any prime ideal Q such that O_{χ}/Q ≃ 𝔽_{q}, χ mod Q is a character of G_{0} with values in 𝔽_{q}^{∗}. In view of the lemma, a simple argument shows that if dim_{ℂ}(H_{χ}) were finite, then dim_{𝔽_{q}}(H_{χ mod Q}) would also be finite. Since there are infinitely many such Q’s, theorem B implies theorem A.

In the spirit of the introduction, we prove theorem B by showing that the set {(σ · π(H(a)))_{σ∈G_{0}} | a ∈ L_{p}} is just as large as it can be, in view of the restriction imposed by the known geometrical relations among the conjugates of the H(a)’s. More precisely, there is a subgroup G_{1} of G_{0} such that G_{1} acts “geometrically” on the involved CM points, whereas the action of the remaining part G_{0}/G_{1} is “chaotic”.

We start with a prime number q ∤ φ(Nd_{K})η and a character χ : G_{0} → 𝔽_{q}^{∗}.

4.1 The geometric part

Let

G_{1} = ⟨ Frob_{Q}(K[p^{∞}]/K)| Q^{2} = q′O_{K}, q′ | d_{K}, q′ ≠ p⟩ ⊂ G_{0}

be the group generated in Gal(K[p^{∞}]/K) by the Frobeniuses of the ramified primes Q_{1}, …, Q_{g} of K that do not divide p. Since these Frobeniuses have order 2, G_{1} is indeed a subgroup of G_{0} and class field theory shows in fact that Frob_{Q_{1}}(K[p^{∞}]/K), …, Frob_{Q_{g}}(K[p^{∞}]/K) is an 𝔽_{2}-base of G_{1}. Let q_{1}, …, q_{g} be the corresponding rational primes and put M = q_{1} ⋯ q_{g}, so that M is prime to pN since all primes dividing N split in K. For d | M, define τ_{d} = ∏_{q_{i}|d} Frob_{Q_{i}}(K[p^{∞}]/K), so that G_{1} = {τ_{d}; d | M}.

There is a unique cyclic subgroup C_{M} ⊂ E(ℂ) of order M which is stable by O_{K}, namely C_{M} = E[Q_{1} ⋯ Q_{g}]. Put C′ = C ⊕ C_{M}, so that C′ ⊂ E(ℂ) is a cyclic subgroup of order NM. Let H′ be the associated family of Heegner points (see section 1, especially Proposition 1.2), so that

∀a ∈ L_{p} : H′(a) = [E/X_{a} → E/X_{a} ⊕ C ⊕ C_{M}] ∈ X_{0}(NM)(K[p^{∞}]).

For d | M, let β_{d} : X_{0}(NM) → X_{0}(N) be the degeneracy map induced by

β_{d}[E → E/C] = [ (E/C[d]) → (E/C[Nd]) ],

for an elliptic curve E with a cyclic subgroup C of order NM.

Lemma 4.2 For all a ∈ L_{p}, β_{d}(H′(a)) = τ_{d} · H(a) ∈ X_{0}(N)(K[p^{∞}]).

Proof: Let c(a) = p^{n} and define Q_{d} = ∏_{q_{i}|d} Q_{i} and Q_{d,n} = Q_{d} ∩ O_{p^{n}}. Then

Q_{d,n} is a proper O_{p^{n}}-ideal and τ_{d} |_{K[p^{n}]}= (\frac{K[p^{n}]/K}{Q_{d,n}} ), so that by proposition 1.3:

τ_{d} · H(a) = [(E/X_{a})^{Q_{d,n}} → (E/X_{a} ⊕ C)^{Q_{d,n}}] .

The inclusion Q_{d,n} ↪ O_{p^{n}} yields a commutative diagram:

(E/X_{a})[Q_{d,n}]  ↪  E/X_{a}  ↠  (E/X_{a})^{Q_{d,n}}
↓  ↓  ↓
(E/X_{a} ⊕ C)[Q_{d,n}] ↪ E/X_{a} ⊕ C ↠ (E/X_{a} ⊕ C)^{Q_{d,n}}

An easy computation shows that

(E/X_{a})[Q_{d,n}] = (X_{a} ⊕ E[Q_{d}])/X_{a} = (X_{a} ⊕ C_{M}[d])/X_{a}.

Similarly, (E/X_{a} ⊕ C)[Q_{d,n}] = (X_{a} ⊕ C ⊕ C_{M}[d])/(X_{a} ⊕ C). It follows that

τ_{d} · H(a) = [E/(X_{a} ⊕ C_{M}[d]) → E/(X_{a} ⊕ C ⊕ C_{M}[d])],

hence indeed τ_{d} · H(a) = β_{d}(H′(a)). □

The restricted character χ : G_{1} → {±1} ⊂ 𝔽_{q}^{∗} lifts to a character with values in {±1} ⊂ ℤ. We can then define the following ℚ-morphisms:

• u : X_{0}(NM) → X_{0}(N)^{2^{g}}, given by u(x) = (β_{d}(x))_{d|M},

• s_{χ} : 𝔸^{2^{g}} → 𝔸, given by s_{χ}(x_{d})_{d|M} = ∑_{d|M} χ (τ_{d}) x_{d},

• π_{χ} : X_{0}(NM) → 𝔸, given by π_{χ} = s_{χ} ∘ (π)^{2^{g}} ∘ u.

Pick a set of representatives {1} ∈ R ⊂ G_{0} of G_{0}/G_{1}. Lemma 4.2 then implies:

e_{χ}(π(H(a)) ⊗ 1) = ∑_{σ∈R} χ^{−1}(σ) · σ · (π_{χ}(H′(a)) ⊗ 1)
= ∑_{σ∈R} χ^{−1}(σ) · (π_{χ}(σ · H′(a)) ⊗ 1).  (1)

Since β_{d} : X_{0}(NM) → X_{0}(N) maps the cusp ∞ ∈ X_{0}(NM) to the cusp ∞ ∈ X_{0}(N), (β_{d})_{∗} : J_{0}(NM) → J_{0}(N) commutes with the usual embeddings of the modular curves into their Jacobians. Let u_{∗} : J_{0}(NM) → J_{0}(N)^{2^{g}} be the product of these maps, and call again s_{χ} : J_{0}(N)^{2^{g}} → J_{0}(N) the map defined by (x_{d})_{d|M} ↦ ∑_{d|M} χ(τ_{d})x_{d}. Put α_{χ} = α ∘ s_{χ} ∘ u_{∗} : J_{0}(NM) → 𝔸, so that α_{χ}(x − ∞) = π_{χ}(x) for all x ∈ X_{0}(NM).

Its dual morphism α_{χ}^{dual} : 𝔸^{dual} → J_{0}(NM)^{dual} = J_{0}(NM) decomposes in:

• α^{dual} : 𝔸^{dual} → J_{0}(N), whose kernel is finite and isomorphic to the 𝔾_{m}- dual of the group of connected components of ker(α).

• s_{χ}^{dual} : J_{0}(N) → J_{0}(N)^{2^{g}}, which is the embedding x ↦ (χ(τ_{d})x)_{d|M}.

• u^{dual} : J_{0}(N)^{2^{g}} → J_{0}(NM), given by (x_{d})_{d|M} ↦ ∑_{d|M} β_{d}^{∗}(x).

Lemma 4.3 The kernel of u^{dual} is finite and its rank divides a power of φ(NM).

Proof: If g = 0, there is nothing to prove. If g ≥ 1, then we can split the morphism u^{dual} into

(J_{0}(N)^{2} ⟶^{v} J_{0}(Nq_{1}))^{2^{g−1}} and J_{0}(Nq_{1})^{2^{g−1}} ⟶^{w} J_{0}(NM),

where v is the sum of the two degeneracy maps J_{0}(N) → J_{0}(Nq_{1}), and w is the analog of u^{dual}, with (N, M) replaced by (Nq_{1}, M/q_{1}). According to [15, Theorem 4.3]:

ker(v) = {(x, y) ∈ Sh_{N} | x + y = 0},

where Sh_{N} = ker(J_{0}(N) → J_{1}(N)) is the Shimura subgroup of J_{0}(N). The lemma follows by induction, since the order of Sh_{N} divides φ(N) [9]. □

If C is the connected component of ker(α_{χ}) and D = J_{0}(NM)/C, we thus obtain exact sequences of proper commutative group schemes over ℚ:

0 → C ⟶^{i} J_{0}(NM) ⟶^{b} D → 0 and 0 → Y → D ⟶^{a} 𝔸 → 0  (2)

such that a ∘ b = α_{χ}, and Y is the Cartier dual of ker(α_{χ}^{dual}), hence finite of rank r, with r dividing a power of φ(Nd_{K})η.

Extending α_{χ} : J_{0}(NM) → 𝔸 to a (surjective) morphism between the Néron models J_{0}(NM)_{/ℤ[1/NM]} = Pic^{0}(X_{0}(NM)_{/ℤ[1/NM]}) and 𝔸_{/ℤ[1/NM]}, we have:

Proposition 4.4 For any rational prime ℓ ∤ 2NM, let J_{0}^{ss}(NM) (𝔽_{ℓ^{2}}) be the subgroup of J_{0}(NM) (𝔽_{ℓ^{2}}) generated by (x − y), for all x, y ∈ X_{0}^{ss}(NM) (𝔽_{ℓ^{2}}). Then

(α_{χ} ⊗ 1)(J_{0}^{ss}(NM) (𝔽_{ℓ^{2}}) ⊗ 𝔽_{q}) = 𝔸 (𝔽_{ℓ^{2}}) ⊗ 𝔽_{q}.

Proof: It follows from a theorem of Ihara [7, Corollary 1, p. 169] that the index of the subgroup J_{0}^{ss}(NM)(𝔽_{ℓ^{2}}) of J_{0}(NM)(𝔽_{ℓ^{2}}) equals the order of the Shimura subgroup Sh_{NM} [13, Proposition 3.6], so that in particular

J_{0}^{ss}(NM) (𝔽_{ℓ^{2}}) ⊗ 𝔽_{q} = J_{0}(NM) (𝔽_{ℓ^{2}}) ⊗ 𝔽_{q}

since q does not divide φ(NM).

The morphisms i, b and a of (2) extend to morphisms between the Néron models of C, J_{0}(NM), D and 𝔸 over ℤ_{ℓ} (which we denote by the same letters); let Y_{/ℤ_{ℓ}} be the kernel of the extended a. Since ℓ ∤ NM, J_{0}(NM), 𝔸, C and D are abelian schemes over ℤ_{ℓ}; since moreover ℓ ≠ 2, the exact sequences of (2) yield fppf-exact sequences of proper commutative group schemes over ℤ_{ℓ} [2, Chap. 7], and Y_{/ℤ_{ℓ}} is a finite flat commutative group scheme of rank r. From these exact sequences, restricted to the special fiber, we get using Lang’s theorem on the triviality of the first Galois cohomology of a connected group over a finite field:

J_{0}(NM)(𝔽_{ℓ^{2}}) → D(𝔽_{ℓ^{2}}) → 0 and D(𝔽_{ℓ^{2}}) → 𝔸(𝔽_{ℓ^{2}}) → H^{1}(𝔽_{ℓ^{2}}, Y) → 0.

Since q does not divide r and r kills H^{1}(𝔽_{ℓ^{2}}, Y), we obtain

(α_{χ} ⊗ 1)(J_{0}(NM) (𝔽_{ℓ^{2}}) ⊗ 𝔽_{q}) = 𝔸 (𝔽_{ℓ^{2}}) ⊗ 𝔽_{q}.

The proposition follows. □

4.2 The chaotic part

Lemma 4.5 [19, Prop. 3.7] ∀(σ ≠ σ′) ∈ R^{2}, σ^{−1}σ′ ∉ [K[p^{∞}]/K, K̂^{(p)∗}].

Proof: We must show that if λ̂ = (λ̂_{q})_{q} ∈ K̂^{(p)∗} (i.e. λ̂_{p} = 1), then

[K[p^{∞}]/K, λ̂] = σ ∈ G_{0} ⟹ σ ∈ G_{1}.

Since the kernel of [K[∞]/K, ⋆] : K̂^{∗} → Gal(K[∞]/K) contains ℚ̂^{∗}K^{∗}, we may also assume that λ̂_{q} ∈ (O_{K})_{q} for all q. Then the proper O_{p^{n}}-ideal I_{n} = O_{p^{n}} λ̂ is integral, and its index d = [O_{p^{n}} : I_{n}] is prime to p and independent of n.

Let r be the order of σ ∈ G_{0}. Since σ |_{K[p^{n}]}= (\frac{K[p^{n}]/K}{I_{n}} ), I_{n}^{r} is a principal O_{p^{n}}-ideal, say I_{n}^{r} = O_{p^{n}}x_{n} with x_{n} ∈ O_{p^{n}}. Then I_{0}^{r} = O_{K}I_{n}^{r} = O_{K}x_{n}, so that (x_{n})_{n≥0} takes only a finite number of values. If x = x_{n} for infinitely many n, then x is an element of ⋂_{n≥0} O_{p^{n}} = ℤ, and I_{n}^{r} = O_{p^{n}}x for all n ≥ 0 (since O_{p^{n}}I_{n′} = I_{n} if n ≤ n′).

Let d_{i}, d_{s} and d_{r} be the divisors of d that correspond respectively to inert, split and ramified primes. Recall that the unique factorization theorem holds in the group of proper O_{p^{n}}-ideals whose norm is prime to p. Since p ∤ d, we thus have a decomposition of I_{n} into a product of prime (and proper) O_{p^{n}}-ideals:

I_{n} = ∏_{Q|q|d} Q^{v_{Q}(I_{n})}.

Since I_{n}^{r} = O_{p^{n}}x, the unique factorization theorem implies:

r × v_{Q}(I_{n}) = r × v\overline{Q}(I_{n}) = \{ \begin{matrix}v_{q}(x) & if q | d_{i}d_{s} \\ 2 × v_{q}(x) & if q | d_{r} \end{matrix}

In particular, v(q) = v_{Q}(I_{n}) does not depend on n or Q | q and

I_{n} = ∏_{q|d_{s}d_{i}} q^{v(q)} ∏_{q|d_{r}} Q^{v(q)}.

Therefore,

σ |_{K[p^{n}]}= ∏_{q|d_{r}} (\frac{K[p^{n}]/K}{Q^{v(q)}} ) = ∏_{q|d_{r}} Frob_{Q}(K[p^{n}]/K)^{v(q)},

and σ belongs to G_{1}. □

In other words, we can apply Theorem 3.1 to H′ and R. So let S be a finite set of inert primes ℓ ∤ 2NMp, and choose for each ℓ a place v_{ℓ} of K[p^{∞}] above ℓ. With notations as in Theorem 3.1, the following map is surjective:

RED : L_{p} → ∏_{ℓ∈S} X_{0}^{ss}(NM)(k(ℓ))^{R}
a ↦ (red_{ℓ}(σ · H′(a)))_{σ∈R,ℓ∈S}

Consider the 𝔽_{q}-linear map:

\begin{aligned}R_{S} : & 𝔸(K[p^{∞}]) ⊗ 𝔽_{q} & → & ⨁_{ℓ∈S} 𝔸(k(ℓ)) ⊗ 𝔽_{q} \\ & x ⊗ 1 & ↦ & ⊕_{ℓ∈S}red_{ℓ}(x) ⊗ 1 \end{aligned}

For ℓ ∈ S and x, y ∈ X_{0}^{ss}(NM)(k(ℓ)), pick a, b ∈ L_{p} such that all components of RED(a) coincide with those of RED(b), except at (ℓ, 1) ∈ S × R where red_{ℓ}(H′(a)) = x and red_{ℓ}(H′(b)) = y. Using equation (1), we compute:

R_{S}(e_{χ}(π(H(a)) ⊗ 1)) − R_{S}(e_{χ}(π(H(b)) ⊗ 1))
= (0, ⋯, 0, (π_{χ}(x) − π_{χ}(y)) ⊗ 1, 0, ⋯, 0)
= (0, ⋯, 0, α_{χ}(x − y) ⊗ 1, 0, ⋯, 0).

Proposition 4.4 then implies that R_{S} is surjective on H_{χ}. In particular:

dim_{𝔽_{q}}(H_{χ}) ≥ ∑_{ℓ∈S} dim_{𝔽_{q}} (𝔸(k(ℓ)) ⊗ 𝔽_{q}).  (3)

To conclude the proof of Theorem B, we must show that it is possible to choose the set S in such a way that the r.h.s of (3) is arbitrarily large.

Put L = K(𝔸[q]). Note that L is Galois over ℚ, embed it into ℂ and let τ ∈ Gal(L/ℚ) be the complex conjugation. Consider the set S_{∞} of rational primes ℓ not dividing 2NMd_{K}q, and such that

Frob_{ℓ}(L/ℚ) = [τ] ∈ Gal(L/ℚ).

Then S_{∞} is infinite. Let S be a finite subset of S_{∞}, and for each ℓ ∈ S, pick a prime Q_{ℓ} of L above ℓ such that Frob_{Q_{ℓ}}(L/ℚ) = τ ∈ Gal(L/ℚ). Pick also a place v′_{ℓ} of K[p^{∞}](𝔸[q]) above Q_{ℓ}, let v_{ℓ} be its restriction to K[p^{∞}] and k(ℓ) the residue field. Then ℓ is inert in K, and

dim_{𝔽_{q}} (𝔸(k(ℓ)) ⊗ 𝔽_{q}) = 2 dim(𝔸).

Hence dim_{𝔽_{q}} (H_{χ}) ≥ #S×2 dim(𝔸) by (3). Taking S arbitrarily large, we obtain dim_{𝔽_{q}} (H_{χ}) ≥ ∞.

5 The case where p divides N

As before, let 𝔸_{/ℚ} be a quotient of J_{0}(N) and π : X_{0}(N) → 𝔸 the induced morphism. Fix a prime number p, and write N = N_{0}p^{μ}, with (N_{0}, p) = 1. We assume that the Heegner Hypothesis holds for N_{0} and K, and choose an ideal N_{0} of O_{K} such that O_{K}/N_{0} ≃ ℤ/N_{0}ℤ. Let E_{/Ω} be an elliptic curve with complex multiplication by O_{K}, and put C = E[N_{0}], so that C ≈ ℤ/N_{0}ℤ.

As in section 1, we can “push forward” C through any p^{n}-isogeny E → E′ to obtain a Γ_{0}(N_{0})-structure on E′. However, in order to get a point on X_{0}(N), we need to add some Γ_{0}(p^{μ})-structure on E′. With L as in section 1, let us consider the following indexing set:

L_{p} = {(a_{1}, a_{2}) ∈ L^{2} | ∃n s.t. X_{a_{1}} ≈ ℤ/p^{n}ℤ, X_{a_{1}} ⊂ X_{a_{2}}, X_{a_{2}}/X_{a_{1}} ≈ ℤ/p^{μ}ℤ}

and the associated family of CM-points:

a = (a_{1}, a_{2}) ∈ L_{p} ⟼ H(a) = [E/X_{a_{1}} → E/X_{a_{2}} ⊕ C] ∈ X_{0}(N).

Since both E/X_{a_{1}} and E/X_{a_{2}} ⊕C have complex multiplication by O_{p^{n}} for some n ≥ 0, Proposition 1.2 implies that H(a) ∈ X_{0}(N)(K[p^{∞}]).

We contend that Theorems A and B generalize mutatis-mutandis to this setting. Since the proof is essentially the same, we will just indicate the necessary modifications, leaving the details to the reader.

First, Theorem B follows from Theorem A and Theorem A from the surjectivity statement of Theorem 3.1 in exactly the same way as before. The only changes in the proof of the latter are due to the fact that L_{p} does not have a tree structure any more. In section 3.2.3 we have to fix a base point e ∈ L_{p} (i.e., a Γ_{0}(p^{μ})-structure on E), and replace GL_{2}(ℤ_{p}) by the ad-hoc congruence subgroup in the definition of T_{p}. Our description of X_{0}^{ss}(N)(k(ℓ)) in section 2.3 did not assume anything on p relative to N, and the computation of red_{ℓ}(σ ·H_{a}) in section 3.3 remains unchanged, as well as the reduction from P GL_{2} to P SL_{2} made in Proposition 3.4, once U = P SL_{2}(ℤ_{p}) has been replaced by a suitable congruence subgroup. Theorem 3.1 thus reduces again to the topological statement of Proposition 3.6, whose proof does not require any more changes.

References

[1] Bertolini M., Selmer groups and Heegner points in anticyclotomic ℤ_{p}- extensions. Comp. Math., 99 (1995), pp. 153-182.

[2] Bosch S., Lütkebohmert W. and Raynaud M., Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag (1990).

[3] Cornut C., Réduction de familles de points CM, Thesis, Prépublication de l’Institut de Recherche Mathématique Avancée, Strasbourg (2000).

[4] Cox D.A., Primes of the form x^{2} + ny^{2}. Wiley & Sons (1989).

[5] Gross B.H., Heights and the special values of L-series, Number Theory (H. Kisilevsky and J. Labute, eds), CMS Conference Proceedings, vol. 7, Amer. Math. Soc. (1987), pp. 115-189.

[6] Gross B.H. and Zagier D., Heegner points and derivatives of L-series. Invent. Math., 84 (1986), pp. 225-320.

[7] Ihara Y., On modular curves over finite fields. Discrete subgroups of Lie groups and applications to moduli, Oxford University Press (1975), pp. 161-202.

[8] Kolyvagin V.A., Euler Systems. The Grothendieck Festschrift. Prog. in Math., Boston, Birkhauser (1990).

[9] Ling S. and Oesterlé J., The Shimura subgroup of J_{0}(N). Astérisque 196-197, Société mathématique de France (1991).

[10] Mazur B., Modular Curves and Arithmetic. Proceedings of the International Congress of Mathematicians, Warsaw 1983, PWN (1984), pp. 185-211.

[11] Nekovář J. and Schappacher N., On the asymptotic behaviour of Heegner points. Turkish J. Math., 23 (1999), pp. 549-556.

[12] Nekovář J., On the parity of ranks of Selmer groups II. Comptes Rendus de l’Acad. Sci. Paris, Série I, 332 (2001), No. 2, pp. 99-104.

[13] Prasad D., Ribet’s theorem: Shimura-Taniyama-Weil implies Fermat, in Seminar on Fermat’s Last Theorem, 1993-1994 (V.K. Murty ed.). CMS Conference Proceedings, 17, (1995), pp. 155-177.

[14] Ratner M., Raghunatan’s conjectures for cartesian products of real and padic Lie groups. Duke Math. J., 77 vol. 2 (1995), pp. 275-382.

[15] Ribet K., Congruence Relations between Modular Forms, Proccedings of the International Congress of Mathematicians, Warsaw 1983, PWN (1984), pp. 503-514.

[16] Serre J-P., Complex Multiplication, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds), Academic Press (1967), pp. 292-296.

[17] Serre J-P. and Tate J., Good reduction of abelian varieties. Ann. Math. 88 (1968), pp. 492-517.

[18] Silverman J. H., The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag (1986).

[19] Vatsal V., Uniform distribution of Heegner points. To appear in Inventiones Math.

[20] Vatsal V., Special values of anticylotomic L-functions. To appear in Duke Math J.

[21] Vignéras M-F., Arithmétique des Algèbres de Quaternions. LNM 800, Springer-Verlag (1980).