The anticyclotomic Main Conjecture for elliptic curves at supersingular primes

Henri Darmon Adrian Iovita

October 1, 2007

Abstract

The Main Conjecture of Iwasawa theory for an elliptic curve E over ℚ and the anticyclotomic ℤ_{p}-extension of an imaginary quadratic field K was studied in [BD2], in the case where p is a prime of ordinary reduction for E. Analogous results are formulated, and proved, in the case where p is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by Perrin-Riou [PR2], [PR4], Pollack [Po1], Kurihara [Ku], Kobayashi [Kob], and Iovita-Pollack [IP] are required to handle this case in which many of the simplifying features of the ordinary setting break down.

Contents

1 Introduction 2

2 The plus/minus p-adic L-functions 5 2.1 Modular forms on quaternion algebras . . . . . . . . . . . . . 5 2.2 Rankin L-functions . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Selmer groups 15 3.1 Local conditions at ℓ ≠ p . . . . . . . . . . . . . . . . . . . . . 17 3.2 Local conditions at p . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Generalised Selmer groups . . . . . . . . . . . . . . . . . . . . 25 3.4 Freeness results for Selmer groups . . . . . . . . . . . . . . . . 27 4 Construction of the Euler system 31

5 Proof of the main result 36

1 Introduction

Let E be an elliptic curve over ℚ of conductor N_{0}, and let K be an imaginary quadratic field of discriminant prime to N_{0}. Choose a rational prime p and let K_{∞} denote the anticyclotomic ℤ_{p}-extension of K. To the datum (E, K, p) are associated two kinds of invariants:

1. The twisted special values L(E/K, χ, 1) of the Hasse-Weil L-series of E over K, as χ ranges over the finite-order characters of G_{∞} := Gal(K_{∞}/K). These special values satisfy certain algebraicity and integrality properties. When p is a prime of ordinary reduction for E, they can be conveniently packaged into a p-adic L-function L_{p}(E, K) which belongs to the Iwasawa algebra Λ := ℤ_{p}[[G_{∞}]].

2. The Selmer group Sel(K_{∞}, E_{p^{∞}}) consisting of classes in H^{1}(K_{∞}, E_{p^{∞}}) which are in the images of the local Kummer maps at all places of K_{∞}. This group is a co-finitely generated Λ-module. One of the interesting features of the anticyclotomic setting is that it need not be Λ-cotorsion in general. Let C denote the characteristic power series of the Pontryagin dual of the Selmer group, setting C = 0 if this Pontryagin dual is not torsion over Λ. The invariant C is well-defined up to multiplication by units of Λ.

It is assumed that the discriminant of K is prime to N := pN_{0} so that K determines a factorization

N = pN^{+}N^{−},

where N^{+} is divisible only by primes which are split in K and N^{−} by primes which are inert in K.

Under certain technical assumptions stated in the introduction of [BD2] which will be recalled below, the article [BD2] proves the following result in the direction of the anticyclotomic main conjecture of Iwasawa theory in the ordinary case. (Cf. Theorem 1 of [BD2].)

Theorem 1.1. Assume that N^{−} is the square-free product of an odd number of primes. Assume also that the prime p is ordinary, and that (E, K, p) satisfies the technical hypotheses stated in 1.6 below. Then C divides the p-adic L-function L_{p}(E, K).

Remark 1.2. It follows from results of Vatsal [Va] that L_{p}(E, K) is nonzero under the hypotheses on N made in Theorem 1.1. In particular, this theorem implies that the Selmer group of E over K_{∞} is Λ-co-torsion. By contrast, when N^{−} is the square-free product of an even number of primes, then L_{p}(E, K) vanishes identically. Vatsal’s theorem on the non-triviality of Heegner points and arguments of Kolyvagin can be used to show that the Selmer group of E over K_{∞} has Λ-corank one in this case.

The main goal of the present note is to formulate and prove an analogous result in the case where p is a prime satisfying

a_{p} := p + 1 − #E(ℤ/pℤ) = 0.

This implies that E is supersingular at p, and is in fact equivalent to this statement when p ≥ 5, in light of the Hasse bound |a_{p}| ≤ 2\sqrt{p}. The foundational study of supersingular main conjectures carried out by Perrin-Riou [PR2], [PR4], Pollack [Po1], Kurihara [Ku], Kobayashi [Kob], and Iovita-Pollack [IP] are required to handle this case in which many of the simplifying features of the ordinary setting break down.

1. The special values L(E/K, χ, 1) cannot be interpolated in an obvious way by an element of Λ. Section 2 explains how the construction of the p-adic L-function L_{p}(E, K) presented in Section 1 of [BD2] can be modified, following the ideas of [PR2] and [Po1], by removing the infinitely many “trivial zeroes” that occur at p-power roots of unity. This process yields two p-adic L-functions L_{p}^{+}(E, K) and L_{p}^{−}(E, K) which both belong to Λ and emerge as the appropriate substitutes for the p-adic L-function in the supersingular setting.

2. In tandem with this analytic complication, the Selmer group Sel(K_{∞}, E_{p^{∞}}) is never a co-torsion Λ-module when p is supersingular. Following an idea of Kobayashi [Kob], Section 3 introduces two restricted Selmer groups

Sel_{+}(K_{∞}, E_{p^{∞}}) and Sel_{−}(K_{∞}, E_{p^{∞}})

defined by imposing more stringent local conditions at the prime p. Let C^{+} and C^{−} denote the characteristic power series of the Pontryagin duals of Sel_{+}(K_{∞}, E_{p^{∞}}) and Sel_{−}(K_{∞}, E_{p^{∞}}) respectively. (Here we follow the same conventions as before, whereby the characteristic power series of a non-torsion Λ-module is taken to be 0.)

The main conjecture that we are interested in is formulated in terms of the plus/minus p-adic L-functions and the restricted Selmer groups, as follows:

Conjecture 1.3. Assume that a_{p} = 0. Then the characteristic power series C^{+} and C^{−} generate the same ideal of Λ as the p-adic L-functions L_{p}^{+}(E, K) and L_{p}^{−}(E, K) respectively.

Fix an integer n ≥ 1. A key ingredient in the proof of Theorem 1.1 given in [BD2] is the construction of certain global cohomology classes

κ(ℓ) ∈ lim_{←}_{m}H^{1}(K_{m}, E[p^{n}]),

indexed by rational primes ℓ satisfying suitable properties (the n-admissible primes in the sense of [BD2]). These classes form a kind of Euler system, as spelled out in Sections 4 and 7 of [BD2]. Section 4 of this paper explains how the construction of [BD2] can be modified in the supersingular case to yield classes κ^{+}(ℓ) and κ^{−}(ℓ) satisfying properties analogous to the classes κ(ℓ) of [BD2]. The strategy of the proof of Theorem 1.1 carries over to establish one of the divisibilities predicted by Conjecture 1.3, which is the main result of this paper.

Theorem 1.4. Assume that a_{p} = 0 and that N^{−} is the square-free product of an odd number of primes. Assume also that (E, K, p) satisfies the hypotheses stated in 1.6 and 1.7 below. Then the characteristic power series C^{+} and C^{−} divide the p-adic L-functions L_{p}^{+}(E, K) and L_{p}^{−}(E, K) respectively.

Remark 1.5. When N^{−} is the square-free product of an even number of primes, the p-adic L-functions L_{p}^{+}(E, K) and L_{p}^{−}(E, K) vanish identically, much as in the ordinary case, and the corresponding Selmer groups are not Λ-co-torsion.

Throughout this article, the following assumptions are made on (E, K, p):

Assumptions 1.6. 1. The prime p is greater or equal to 5.

2. The Galois representation attached to E_{p} has image isomorphic to GL_{2}(𝔽_{p}).

3. There is a modular parameterization X_{0}(N_{0}) ⟶ E whose degree is not divisible by p.

4. For all primes ℓ such that ℓ^{2} divides N, and p divides ℓ + 1, the module E_{p} is an irreducible I_{ℓ}-module, where I_{ℓ} denotes the inertia group at ℓ.

These assumptions are made mainly to simplify the arguments and could probably be relaxed at the cost of complicating the proofs. (See Remark 1 after the statement of Assumption 6 in the introduction of [BD2].) The next set of assumptions, which does not appear in [BD2], is imposed on us by our lack of understanding of the local condition to impose at p in defining the appropriate Selmer group when p is a supersingular prime and p is inert in K.

Assumptions 1.7. 1. The prime p is split in K, so that it can be written p = 𝔭𝔭̄, where 𝔭 is a prime of K.

2. The prime 𝔭 is totally ramified in the anticyclotomic ℤ_{p}-extension attached to K.

Assumption 2 is automatically satisfied if p does not divide the class number of K. It would be desirable to be able to dispense with assumption 1 and treat the inert primes on the same footing as the split primes, as is done in [BD2] when p is ordinary. Acknowledgements: The authors are grateful to the anonymous referee for a careful proofreading which led to corrections of some inaccuracies and to significant improvements in the presentation. Both authors were supported by grants from NSERC. The first author was supported by a James McGill Chair, and the second by a Canada Research Chair, while this research was conducted.

2 The plus/minus p-adic L-functions

2.1 Modular forms on quaternion algebras

Let N^{+} and N^{−} be positive integers such that N^{+} is divisible only by primes which are split in K and N^{−} by primes which are inert in K. Assume that N^{−} is square-free, and has an odd number of prime factors. Let p be a prime which does not divide N_{0} := N^{+}N^{−}.

Let B be the definite quaternion algebra of discriminant N^{−}∞, and let R be an Eichler ℤ[\frac{1}{p}]-order of conductor N^{+} in B. Since p does not divide N^{−}, we may fix an isomorphism

ι : B_{p} := B ⊗ ℚ_{p} ⟶ M_{2}(ℚ_{p}).

Let R_{1}^{×} denote the group of elements of reduced norm one in R, and define

Γ = ι(R^{×}) ⊂ GL_{2}(ℚ_{p}).

Let T denote the Bruhat-Tits tree attached to PGL_{2}(ℚ_{p}), whose set V(T ) of vertices is in bijection with the similarity classes of ℤ_{p}-lattices in ℚ_{p}^{2}, two vertices being joined by an edge if the corresponding classes of lattices admit representatives in which one contains the other with index p. The group GL_{2}(ℚ_{p}) (and hence, in particular, Γ) acts naturally on T and the quotient of T by the action of Γ is a finite graph.

Definition 2.1. A modular form of weight 2 on V(T ) for Γ is a Γ-invariant ℤ_{p}-valued function on V(T ).

Denote by S_{2}(V/Γ) the space of all such forms; it is a finitely generated ℤ_{p}-module equipped with a natural action of the Hecke operators T_{ℓ} (with ℓ |̸N) described as in [BD2], Section 1.1. Furthermore, it admits a natural ℤ-structure S_{2}(V/Γ)^{ℤ} consisting of the ℤ-valued functions in S_{2}(V/Γ), which is preserved by the action of the Hecke operators.

Let f_{E} ∈ S_{2}(Γ_{0}(N_{0}))^{new} be the eigenform of weight two corresponding to E. For each prime ℓ not dividing N we have

T_{ℓ}(f_{E}) = a_{ℓ}(E)f_{E},

where T_{ℓ} denotes the Hecke operator acting on the space of classical cusp forms on Γ_{0}(N_{0}).

Theorem 2.2 (Jacquet-Langlands). There is an eigenform f ∈ S_{2}(V/Γ)^{ℤ} with the same Hecke eigenvalues as those attached to f_{E}, i.e., such that

T_{ℓ}(f) = a_{ℓ}(E)f, for all ℓ |̸N.

This form is unique up to multiplication by a non-zero scalar.

In addition, the function f satisfies the following property:

Proposition 2.3. For all v ∈ V(T ),

∑_{v′↔v} f(v′) = a_{p}(E)f(v),

where the sum is taken over the p + 1 vertices v′ adjacent to v.

Proof. This follows directly from the description of the action of the Hecke operator T_{p} on S_{2}(V/Γ):

(T_{p}f)(v) = ∑_{v′↔v} f(v′),

and the fact that f is an eigenvector for T_{p} with associated eigenvalue a_{p}(E).

We normalize the form f so that it is not divisible by any integer in S_{2}(V/Γ)^{ℤ}. This makes f well-defined up to a sign. As shall be seen in the next section, the p-adic L-function attached to E and K is defined directly in terms of f rather than in terms of the classical cusp form f_{E}.

2.2 Rankin L-functions

The goal of this section is to define a p-adic L-function attached to a modular form f ∈ S_{2}(V/Γ) satisfying a_{p}(f) = 0 and to a quadratic subfield K ⊂ B, by combining the construction described in Section 1.2. of [BD2] with the ideas of Pollack [Po1].

Suppose for simplicity that the discriminant of K is prime to N. Let O_{K} denote the ring of integers of K and let O := O_{K}[1/p] denote its ring of {p}-integers. For the sake of concreteness, we present the construction under the further assumption that the class number of O_{K}[1/p] is equal to 1. The reader may, if she wishes, adapt the construction to general class number by following the approach described in Section 1.2 of [BD2]. The advantage of the class number one assumption is that it allows for a different, more concrete and geometric—purely p-adic, instead of adelic—presentation of Section 1.2 of [BD2], enabling the authors to avoid what would otherwise be a somewhat tedious repetition of the constructions in that section.

Let n be fixed integer, and let f be an eigenform in S_{2}(V/Γ) satisfying a_{p}(f) ≡ 0 (mod p^{n}). Let

Ψ : K ⟶ B

be an embedding of algebras, satisfying

Ψ(K) ∩ R := Ψ(O).

Such an embedding exists if and only if all the primes dividing N^{+} are split in K, while those dividing N^{−} are inert in K. It is then unique, up to conjugation by the action of R^{×}, thanks to the class number one assumption. Let K_{p} := K ⊗ ℚ_{p} be the p-adic completion of K. The embedding Ψ induces an action of the p-adic group Π_{∞} := K_{p}^{×}/ℚ_{p}^{×} on T by isometries by setting

g ⋆ x := ιΨ(g)(x),  (1)

for any g ∈ K_{p}^{×} and x any vertex or edge of T . We begin by defining certain partial p-adic L-functions attached to Ψ, by studying this action. It is somewhat clearer to separate the study into two cases: Case 1: Suppose p is split in K. The choice of a prime 𝔭 of K above p induces a homomorphism

| |_{p} : K_{p}^{×}/ℚ_{p}^{×} ⟶ ℤ

defined by

|α|_{p} := ord_{𝔭}(α/ᾱ).

Note that replacing 𝔭 by 𝔭̄ only changes the resulting homomorphism | |_{p} by a sign, so that the abuse of notation inherent in the notation | |_{p} is not serious. The kernel of | |_{p} is the maximal compact subgroup of Π_{∞}, denoted U_{0}. This group is identified with ℤ_{p}^{×} under the map which sends α to α/ᾱ. Let

… ⊂ U_{n} ⊂ … ⊂ U_{1} ⊂ U_{0}  (2)

be the the natural decreasing filtration of the group U_{0} by subgroups in which the index of U_{n} is (p − 1)p^{n−1}. In the action of Π_{∞} on T of (1), the maximal compact subgroup U_{0} fixes a sequence (infinite in both directions) of consecutive vertices, i.e., a geodesic J = J_{Ψ} of T . The quotient Π_{∞}/U_{0} ≅ ℤ acts by translation on this geodesic.

The distance between a vertex v of T and the geodesic J_{Ψ} is defined to be the shortest distance between v and a vertex of J_{Ψ}. If the distance from v to J_{Ψ} is equal to n, then the stabilizer of v in Π_{∞} is exactly U_{n}, and the quotient Π_{∞}/U_{n} acts simply transitively on the set of vertices at distance n from J_{Ψ}.

Now let us fix a sequence of consecutive vertices v_{0}, v_{1}, v_{2}, ... such that v_{n} is at distance n from J = J_{Ψ}. (See Figure 1 for an illustration in the case where p = 2.)

We define a sequence of functions

f_{K,n} : Π_{∞}/U_{n} ⟶ ℤ_{p}, by the rule f_{K,n}(α) = f(α ⋆ v_{n}).

Figure 1: Action of K_{p}^{×} on T when p is split.

Let u_{p} be a fundamental p-unit of K, i.e., a generator of the group of elements in O_{K}[1/p]^{×} of norm one, modulo torsion. The quotient

G̃_{∞} = Π_{∞}/u_{p}^{ℤ}

is a compact p-adic group. By abuse of notation, the groups U_{j} occurring in the filtration (2) and their natural images in G̃_{∞} will be denoted by the same symbol.

Lemma 2.4. The functions f_{K,n} are invariant under translation by u_{p}, and hence descend to functions on G̃_{∞}/U_{n}.

Proof. Note that

f_{K,n}(u_{p}α) = f((u_{p}α) ⋆ v_{n}) = f(ιΨ(u_{p})(α ⋆ v_{n})),

and that ιΨ(u_{p}) belongs to Γ. The result therefore follows from the invariance of f under translation by elements of Γ.

Thanks to Lemma 2.4, the functions f_{K,n} defined above can be viewed as functions on the finite quotients G̃_{∞}/U_{n}. Case 2: Suppose now that p is inert in K. This case is somewhat simpler, because Π_{∞} = K_{p}^{×}/ℚ_{p}^{×} is already compact, and is equal, therefore, to its maximal compact subgroup U_{0}. This group is identified with the group of elements in O_{K} ⊗ ℤ_{p} of norm one under the map which sends α to α/ᾱ. Let

… ⊂ U_{n} ⊂ … ⊂ U_{1} ⊂ U_{0} = Π_{∞}  (3)

Figure 2: Action of K_{p}^{×} on T when p is inert.

be the the natural decreasing filtration of the group U_{0} by subgroups of index (p + 1)p^{n−1}. The group G̃_{∞} fixes a distinguished vertex v_{0}. If the distance from a vertex v to v_{0} is equal to n, then the stabilizer of v in G̃_{∞} is exactly U_{n}, and the quotient G̃_{∞}/U_{n} acts simply transitively on the set of vertices at distance n from v_{0}.

Fix a sequence of consecutive vertices v_{0}, v_{1}, v_{2}, … such that v_{n} is at distance n from v_{0}. (See Figure 2).

We then define a sequence of functions

f_{K,n} : G̃_{∞}/U_{n} ⟶ ℤ_{p}, by the rule f_{K,n}(α) = f(α ⋆ v_{n}).

In conclusion, in both the cases where p is split or inert in K, we have associated to f and Ψ a sequence of functions

f_{K,n} : G̃_{n} ⟶ ℤ_{p}, where G̃_{n} := G̃_{∞}/U_{n}.

We associate to these functions a sequence of elements L̃_{n} ∈ ℤ_{p}[G̃_{n}] by setting:

L̃_{n} := ∑_{σ∈G̃_{n}} f_{K,n}(σ)σ^{−1} ∈ ℤ_{p}[G̃_{n}].

Remark 2.5. The assumption that the class number of O is equal to 1 can readily be disposed of following the treatment given in Section 1.2 of [BD2].

The definitions given there would yield a sequence L̃_{n} as above, belonging to the finite group rings ℤ_{p}[G̃_{n}], where now G̃_{∞} denotes the group

G̃_{∞} := (K ⊗ Ẑ)^{×}/ ((ℚ ⊗ Ẑ)^{×} ∏_{ℓ≠p}(O ⊗ ℤ_{ℓ})^{×}K^{×}) ,

which is identified with the Galois group over K of the union of all ring class fields of K of p-power conductor, an extension which contains the Hilbert class field of K.

Denote by

π_{n+1,n} : ℤ_{p}[G̃_{n+1}] ⟶ ℤ_{p}[G̃_{n}]

the ring homomorphism induced by the natural projection G̃_{n+1} ⟶ G̃_{n}. Occasionally we will abuse notation and view L̃_{n} as an element of ℤ_{p}[G̃_{n+1}] by replacing it by an arbitrary lift to this ring under the homomorphism π_{n+1,n}. Of course this element is not well-defined, but the product

ξ̃_{n}L̃_{n} ∈ ℤ_{p}[G̃_{n+1}]

is well-defined, where

ξ̃_{n} = ∑_{s∈U_{n}/U_{n+1}} s.

Lemma 2.6. The elements L̃_{n} satisfy the following compatibility relations under the projections π_{n+1,n}:

π_{n+1,n}(L̃_{n+1}) = a_{p}(E)L̃_{n} − ξ_{n−1}L̃_{n−1}.

In particular, if p is supersingular for E so that a_{p}(E) = 0,

π_{n+1,n}(L̃_{n+1}) = −ξ̃_{n−1}L̃_{n−1}, for all n ≥ 1.

Proof. If g_{n} is any element of G̃_{n}, let g_{n+1} denote an arbitrary lift of this element to G̃_{n+1}. A direct calculation shows that

π_{n+1,n}(L̃_{n+1}) = ∑_{g_{n}∈G̃_{n}} ( ∑_{s∈U_{n}/U_{n+1}} f_{K,n+1}(sg_{n+1})) g_{n}^{−1}.  (4)

Figure 3: The inner sum

On the other hand,

∑_{s∈U_{n}/U_{n+1}} f_{K,n+1}(sg_{n+1}) = ∑_{s∈U_{n}/U_{n+1}} f((sg_{n+1}) ⋆ v_{n+1}).  (5)

The sum on the right corresponds to summing the function f over the p vertices which are adjacent to the vertex g_{n}⋆v_{n} and are different from g_{n}⋆v_{n−1} (see figure 3).

It follows from proposition 2.3 satisfied by f that

∑_{s∈U_{n}/U_{n+1}} f((sg_{n+1}) ⋆ v_{n+1}) = a_{p}(E)f(g_{n} ⋆ v_{n}) − f(g_{n} ⋆ v_{n−1}) (6)
= a_{p}(E)f_{K,n}(g_{n}) − f_{K,n−1}(g_{n}).  (7)

The lemma follows by combining (4), (5), and (7).

We may write

G̃_{∞} = Δ × G_{∞},

where Δ is the torsion subgroup of G̃_{∞} and G_{∞} is its maximal torsion-free quotient, which is topologically isomorphic to ℤ_{p}. The image of Δ in G̃_{n} is identified with Δ, and the group G_{∞} can be written as

G_{∞} = lim_{←} G_{n}, where G_{n} := G̃_{n+1}/Δ ≃ ℤ/p^{n}ℤ.

Let π : ℤ_{p}[G̃_{n+1}] ⟶ ℤ_{p}[G_{n}] be the natural homomorphism induced by the projection G̃_{n+1} ⟶ G_{n}. Then for each n ≥ 0, we set:

L_{n} := π(L̃_{n+1}).

By abuse of notation, denote again by

π_{n+1,n} : ℤ_{p}[G_{n+1}] ⟶ ℤ_{p}[G_{n}]

the maps induced by the natural group homomorphisms. The elements L_{n} inherit from the L̃_{n} the compatibility properties of Lemma 2.6 under these maps, i.e., when p is supersingular:

π_{n+1,n}(L_{n+1}) = −ξ_{n}L_{n−1}, for all n ≥ 1.  (8)

Here ξ_{n} := π(ξ̃_{n+1}) is the element of ℤ_{p}[G_{n}] given by ξ_{n} = ∑_{σ∈H_{n}} σ, where

H_{n} = ker(G_{n} ⟶ G_{n−1}).

Let us fix a topological generator

γ ∈ lim_{←} G_{n} ≅ ℤ_{p}.

This determines the identification sending γ to 1 + T

Λ = lim_{←} ℤ_{p}[G_{n}] ≅ ℤ_{p}[[T]].

In this way ℤ_{p}[G_{n}] is identified with ℤ_{p}[T]/ω_{n}ℤ_{p}[T], where ω_{n} = (T + 1)^{p^{n}} − 1, and the element ξ_{n} is identified with the p^{n}-power cyclotomic polynomial in T + 1. (In other words, the roots of ξ_{n}(T) are of the form ζ − 1, where ζ ranges over all primitive p^{n}-th roots of unity.) Note that we have

ω_{n}(T) = T ∏_{j=1}^{n} ξ_{j}(T).

Let us also write

ω̃_{n}^{+}(T) := ∏_{j even^{j=2}}^{n} ξ_{j}(T), ω̃_{n}^{−}(T) := ∏_{j odd^{j=1}}^{n} ξ_{j}(T),

and set ω_{n}^{±}(T) = T ω̃_{n}^{±}(T). The following technical lemma is straightforward to derive, but we note it for better reference.

Lemma 2.7. Let n be a positive integer and ϵ denote the sign of (−1)^{n}.

1. Multiplication by ω̃_{n}^{−ϵ} induces a natural isomorphism

Λ/(ω_{n}^{ϵ}) ⟶ ω̃_{n}^{−ϵ}Λ/(ω_{n}).

2. For all r ≥ 1, multiplication by ω̃_{n}^{−ϵ} induces a natural isomorphism

Λ/(ω_{n}^{ϵ}, p^{r}) ⟶ ω̃_{n}^{−ϵ}Λ/(ω_{n}, p^{r}).

3. If X is a free Λ_{r,n} := Λ/(ω_{n}, p^{r})-module and x ∈ X is annihilated by ω_{n}^{ϵ}, then there is a unique y ∈ X/ω_{n}^{ϵ}X such that x = ω̃_{n}^{−ϵ}y.

The following proposition is key in the construction of the plus and minus p-adic L-functions.

Proposition 2.8. Let ϵ denote the sign of (−1)^{n}. Then

1. ω_{n}^{ϵ}L_{n} = 0

2. There is a unique element L_{n}^{ϵ} ∈ Λ/ω_{n}^{ϵ}Λ such that L_{n} = ω̃_{n}^{−ϵ}L_{n}^{ϵ}.

Proof. For the first assertion, suppose first that n > 2 is even. Then

ω_{n}^{+}L_{n} = ω_{n−2}^{+}ξ_{n}L_{n} = ω_{n−2}^{+}ξ_{n}π_{n,n−1}(L_{n}).

But by equation (8),

ω_{n−2}^{+}ξ_{n}π_{n,n−1}(L_{n}) = −ω_{n−2}^{+}ξ_{n}ξ_{n−2}L_{n−2}.

This allows the statement to be reduced by induction to the case n = 2. For this value of n it follows from the direct calculation

ω_{2}^{+}L_{2} = T ξ_{2}(T)L_{2} = T ξ_{2}π_{2,1}(L_{2}) = −T ξ_{1}ξ_{2}L_{0},

where the last equality follows from (8). But this expression is 0 because T ξ_{1}ξ_{2} = 0 in ℤ_{p}[G_{2}]. The proof when n is odd is identical. For the second (divisibility) assertion, we invoke Lemma 2.7, noting that we have ω_{n} = ω_{n}^{ϵ}ω̃_{n}^{−ϵ}. Therefore an element of Λ/ω_{n}Λ annihilated by ω_{n}^{ϵ} is divisible by ω̃_{n}^{−ϵ} and the result of the division is unique in Λ/ω_{n}^{ϵ}Λ.

Let us now denote by

\{ \begin{matrix}L_{n}^{+} = (−1)^{\frac{n}{2}} L_{n}^{+} & if n is even ; \\ L_{n}^{−} = (−1)^{\frac{n+1}{2}} L_{n}^{−} & if n is odd. \end{matrix}

Lemma 2.9. The sequence {L_{n}^{+}}_{n even} is compatible with respect to the natural projections

Λ/ω_{n}^{+} ⟶ Λ/ω_{n−2}^{+},

and likewise for the sequence {L_{n}^{−}}_{n odd}.

Proof. Let us, for all n ≥ 0, choose lifts of L_{n} and L_{n}^{±} to Λ and denote them by the same symbols. Suppose first that n ≥ 2 is even. Then we have

L_{n} = −ξ_{n−1}L_{n−2} (mod ω_{n−1}).

This implies that there exists F ∈ Λ such that

ω̃_{n}^{−}L_{n}^{+} = −ξ_{n−1}ω̃_{n−2}^{−}L_{n−2}^{+} + ω_{n−1}F.

Noting that ω_{n−1} = ω_{n−2}^{+}ω̃_{n}^{−}, we can cancel by ω̃_{n}^{−} = ξ_{n−1}ω̃_{n−2} to obtain

L_{n}^{+} = −L_{n−2}^{+} + ω_{n−2}^{+}F,

which proves the statement when n is even. The case where n is odd is similar.

Thanks to this lemma we may denote by

L_{f}^{+} := lim_{←} L_{n}^{+} ∈ lim_{←} Λ/ω_{n}^{+} ≅ Λ,

and define L_{f}^{−} ∈ Λ similarly. Let L ↦ L^{∗} denote the involution in Λ sending every group- like element in this completed group ring to its inverse. We set

L_{p}(f, K)^{±} := L_{f}^{±}(L_{f}^{±})^{∗},

following definition 1.6 of [BD2].

3 Selmer groups

Class field theory identifies G̃_{∞} with Gal(K̃_{∞}/K), where K̃_{∞} is the union of all the ring class fields of K of p-power conductor. For each integer m ≥ 0, the quotient G̃_{m} is identified with Gal(K̃_{m}/K), where K̃_{m} is the ring class field of K of conductor p^{m}. The subfield of K̃_{∞} fixed by Δ is the anticyclotomic ℤ_{p}-extension K_{∞} of K, so that

G_{∞} = Gal(K_{∞}/K).

Under this identification, the group G_{m} corresponds to the Galois group Gal(K_{m}/K), where K_{m} is defined to be the m-th layer of the ℤ_{p}-tower K_{∞}.

Following the notations of Section 2.1 of [BD2], denote by V_{f} the twodimensional Galois representation attached to the modular form f (with coefficients in ℚ_{p}) and let T_{f} denote a G_{ℚ} stable ℤ_{p}-sub-lattice. The finite modules

T_{f,n} := T_{f}/p^{n}T_{f} = (V_{f}/T_{f})[p^{n}] (n ≥ 1)

fit naturally both into a projective and an inductive system, i.e., for all r ≥ n there are natural maps

T_{f,r} ↠ T_{f,n}, T_{f,n} ↪ T_{f,r},

which will be used to take both projective and injective limits of cohomology groups associated to the T_{f,n}.

The main goal of this section is to define certain Selmer groups attached to the Galois representations T_{f,n}, and to prove certain basic facts about their structure. For each m ≥ 0 and n ≥ 1, the Selmer group Sel(K_{m}, T_{f,n}) is defined as a subgroup of the (continuous) Galois cohomology group H^{1}(K_{m}, T_{f,n}) by imposing conditions on restrictions to local decomposition groups.

More precisely, for every rational prime ℓ, let

K_{m,ℓ} := K_{m} ⊗_{ℚ} ℚ_{ℓ} = ⊕_{λ|ℓ}K_{m,λ}, H^{1}(K_{m,ℓ}, T_{f,n}) := ⊕_{λ|ℓ}H^{1}(K_{m,λ}, T_{f,n}).

There is a natural restriction map

res_{ℓ} : H^{1}(K_{m}, T_{f,n}) ⟶ H^{1}(K_{m,ℓ}, T_{f,n}).

For each rational prime ℓ we define certain distinguished subgroups

H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) ⊂ H^{1}(K_{m,ℓ}, T_{f,n}),

referred to as the finite part of these local cohomology groups as follows. Let A_{f} denote the abelian variety associated to the modular form f, as described in greater detail in Section 4. Let V_{p}(A_{f}) := T_{p}(A_{f}) ⊗ ℚ_{p} be the p-adic Galois representation associated to A_{f}, and let H_{fin}^{1}(K_{m,ℓ}, V_{p}(A_{f})) denote the image of A_{f}(K_{m,ℓ}) ⊗ ℤ_{p} in H^{1}(K_{m,ℓ}, V_{p}(A_{f})) under the Kummer map. By construction, the two-dimensional Galois representation V_{f} is a quotient of V_{p}(A_{f}). Write

π_{f} : V_{p}(A_{f}) ⟶ V_{f}, H^{1}(K_{m,ℓ}, V_{p}(A_{f})) ⟶ H^{1}(K_{m,ℓ}, V_{f})

for the associated G_{ℚ}-equivariant projection, as well as for the maps induced by it on the various local cohomology groups. We define

H_{fin}^{1}(K_{m,ℓ}, V_{f}) := π_{f}(H_{fin}^{1}(K_{m,ℓ}, V_{p}(A_{f}))),

and H_{fin}^{1}(K_{m,ℓ}, T_{f}) as the inverse image of H_{fin}^{1}(K_{m,ℓ}, V_{f}) under the natural map

H^{1}(K_{m,ℓ}, T_{f}) ⟶ H^{1}(K_{m,ℓ}, V_{f})

induced by the inclusion T_{f} ↪ V_{f}. Finally, for every integer n ≥ 1 we let H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) denote the image of H_{fin}^{1}(K_{m,ℓ}, T_{f}) in H^{1}(K_{m,ℓ}, T_{f,n}) under the map induced by the canonical projection T_{f} ⟶ T_{f,n}.

Definition 3.1. The Selmer group attached to K_{m} and T_{f,n} is the group of cohomology classes s ∈ H^{1}(K_{m}, T_{f,n}) satisfying

res_{ℓ}(s) belongs to H_{fin}^{1}(K_{m,ℓ}, T_{f,n}), for all ℓ.

It is denoted Sel(K_{m}, T_{f,n}).

We also set

Sel(K_{∞}, T_{f,n}) := lim_{m} Sel(K_{m}, T_{f,n}), Sel(K_{∞}, T_{f,∞}) := lim_{n} Sel(K_{∞}, T_{f,n}),

(9)

where the direct limits are taken with respect to the natural maps induced by restriction and the inclusions T_{f,n} ↪ T_{f,n′}.

3.1 Local conditions at ℓ ≠ p

We begin by discussing the groups H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) in the case where ℓ ≠ p.

We define (following [BD2]) the singular part of the local cohomology group H^{1}(K_{m,ℓ}, T_{f,n}) to be the quotient

H_{sing}^{1}(K_{m,ℓ}, T_{f,n}) := \frac{H^{1}(K_{m,ℓ}, T_{f,n})}{H_{fin}^{1}(K_{m,ℓ}, T_{f,n})}.

If ℓ does not divide N, then

H_{sing}^{1}(K_{m,ℓ}, T_{f,n}) = H^{1}(I_{m,ℓ}, T_{f,n})^{G_{K_{ℓ}}} := ∏_{λ} H^{1}(I_{m,λ}, T_{f,n})^{G_{K_{ℓ}}},

where λ runs over the primes of K_{m} over ℓ and I_{m,λ} is the inertia subgroup of G_{K_{m,λ}}.

Restriction defines a so-called residue map

∂_{ℓ} : H^{1}(K_{m,ℓ}, T_{f,n}) ⟶ H_{sing}^{1}(K_{m,ℓ}, T_{f,n})

such that the following sequence is exact

0 ⟶ H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) ⟶ H^{1}(K_{m,ℓ}, T_{f,n}) ⟶ H_{sing}^{1}(K_{m,ℓ}, T_{f,n}).

The following gives a local control theorem for the Selmer group.

Lemma 3.2. For all rational primes ℓ ≠ p, the natural map induced by restriction

H_{sing}^{1}(K_{ℓ}, T_{f,n}) ⟶ H_{sing}^{1}(K_{m,ℓ}, T_{f,n})

is injective.

For example, when ℓ does not divide N, this follows from the fact that K_{m}/K is unramified at the primes above ℓ, so that any class which becomes unramified over K_{m} already had to be unramified over K.

The cup product in local Galois cohomology combined with the Weil pairing T_{f,n} × T_{f,n} ⟶ μ_{p^{n}} leads to the non-degenerate local Tate pairing

H^{1}(K_{m,ℓ}, T_{f,n}) × H^{1}(K_{m,ℓ}, T_{f,n}) ⟶ H^{2}(K_{m,ℓ}, μ_{p^{n}}) ⟶^{inv_{ℓ}} ℤ/p^{n}ℤ,

in which the rightmost map is given by

inv_{ℓ}(κ) := ∑_{λ|ℓ} inv_{λ}(κ),

where inv_{λ} is the standard identification of H^{2}(K_{m,λ}, μ_{p^{n}}) with ℤ/p^{n}ℤ given by local class field theory.

It is a standard fact that the groups H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) are maximal isotropic for the local Tate pairing, and that this pairing therefore induces a perfect duality

H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) × H_{sing}^{1}(K_{m,ℓ}, T_{f,n}) ⟶ ℤ/p^{n}ℤ.

(Cf. Proposition 2.3 of [BD2].)

We now recall the definition of admissible primes given in Section 2.2 of [BD2].

Definition 3.3. A rational prime ℓ is said to n-admissible relative to f if it satisfies the following conditions:

1. ℓ does not divide pN;

2. ℓ is inert in K/ℚ;

3. p does not divide ℓ^{2} − 1;

4. p^{n} divides ℓ + 1 − a_{ℓ} or ℓ + 1 + a_{ℓ}.

One of the motivations for singling out these primes is the following freeness result for the local cohomology group H^{1}(K_{m,ℓ}, T_{f,n}) when ℓ is nadmissible.

Lemma 3.4. If ℓ is an n-admissible prime, then

1. The groups H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) and H_{sing}^{1}(K_{m,ℓ}, T_{f,n}) are free of rank one over Λ_{n,m} = Λ/(p^{n}, ω_{m}).

2. The group H^{1}(K_{m,ℓ}, T_{f,n}) is free of rank two over Λ_{n,m}.

Proof. See Lemma 2.7 of [BD2].

An important caveat that needs to be noted is that the finite part at ℓ does not just depend on the underlying Galois representation T_{f,n}, but on the Galois representation V_{f} from which it arises. For example, if T_{f,n} comes from the p-division points of an abelian variety A_{f} with good reduction at ℓ, then the representation T_{f,n} is of course unramified, and H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) merely consists of the unramified cohomology classes. This is not the case if T_{f,n} is unramified but arises from an abelian variety with multiplicative reduction at ℓ. If ℓ is a prime which divides N exactly, then V_{f} is is an ordinary representation of G_{ℚ_{ℓ}}: it contains a unique one-dimensional ℚ_{p}- vector subspace which is stable under the action of the decomposition group at ℓ. Let T_{f,n}^{(ℓ)} denote the corresponding rank one (ℤ/p^{n}ℤ)-submodule of T_{f,n}. Then H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) is the image in H^{1}(K_{m,ℓ}, T_{f,n}) of H^{1}(K_{m,ℓ}, T_{f,n}^{(ℓ)}).

3.2 Local conditions at p

In the supersingular setting, the classical definition of the Selmer group Sel(K_{∞}, T_{f,∞}) given in (9) suffers from the fact that the resulting object is not a co-torsion module over the Iwasawa algebra Λ = ℤ_{p}[[G_{∞}]]. The idea is to cut down the size of this Selmer group by imposing more stringent local conditions at the primes above p.

We will follow closely Sections 4 and 6 of [IP] with some adjustments due to the fact that here we work with torsion coefficients. In order to define the appropriate subgroups of H_{fin}^{1}(K_{m,p}, T_{f,n}), we need to make the following assumptions which are satisfied in our application:

Assumptions 3.5. 1. The prime p is split in K, so that it can be written p = 𝔭𝔭̄, where 𝔭 is a prime of K.

2. The prime 𝔭 is totally ramified in the anticyclotomic ℤ_{p}-extension K_{∞} attached to K.

3. The Galois representation T_{f,n} is isomorphic (as a representation of G_{ℚ_{p}}) to E[p^{n}], where E is an elliptic curve over ℚ_{p} with supersingular reduction at p.

Note that this implies, in particular, that

a_{p}(f) ≡ 0 (mod p^{n}).

Recall that K_{m} denotes the m-th layer in the anticyclotomic ℤ_{p}-extension K_{∞}. Let K_{m,p} := K_{m} ⊗ ℚ_{p} = K_{m,𝔭} ⊕ K_{m,𝔭̄} denote the completion of K_{m} at p. Let Ê denote the formal group of E_{ℚ_{p}}. First we will recall the description of

Ê(K_{m,p}) := Ê(K_{m,𝔭}) ⊕ Ê(K_{m,𝔭̄})

as a ℤ_{p}[G_{m}] = Λ_{m}-module, for all m ≥ 0.

Since the discussion in this section is purely local, we will lighten notations by letting {L_{m}}_{m≥0} denote either of the following towers of local fields: {K_{m,𝔭}}_{m≥0} or {K_{m,𝔭̄}}_{m≥0}.

The following theorem is essential in defining the plus and minus Selmer groups attached to T_{f,n}.

Theorem 3.6. For m ≥ 0 there exist points d_{m} ∈ Ê(L_{m}) such that 1. Tr_{m−1}^{m}(d_{m}) = −d_{m−2} for all m ≥ 2

2. Tr_{0}^{1}(d_{1}) = ud_{0} with u ∈ ℤ_{p}^{×}

3. d_{m}, d_{m−1} generate Ê(L_{m}) as a ℤ_{p}[G_{m}]-module and d_{0} generates Ê(L_{0}) as a ℤ_{p}-module.

This theorem is proved in [IP]. (See Theorem 4.5 of [IP].)

Using the sequence of points {d_{m}}_{m≥0} we consider two subsequences

d_{m}^{+} = \{ \begin{matrix}d_{m} & if m is even; \\ d_{m−1} & if m is odd; \end{matrix} d_{m}^{−} = \{ \begin{matrix}d_{m−1} & if m ≥ 2 is even; \\ d_{m} & if m is odd. \end{matrix}

Now we define Ê^{±}(L_{m}) := Λ_{m}d_{m}^{±} ⊂ Ê(L_{m}).

Let us remark that the Λ_{m}-submodule Ê^{±}(L_{m}) thus defined is independent of the choice of the sequence of points {d_{m}}_{m≥0} as in Theorem 3.6. (See Lemma 4.13 of [IP].)

Let us now fix integers m, n.

Lemma 3.7. The natural map

j : Ê^{±}(L_{m})/p^{n}Ê^{±}(L_{m}) ⟶ Ê(L_{m})/p^{n}Ê(L_{m})

is injective for all m, n.

Proof. We consider the case where the sign is +, the other case being proved in a similar way. Let P ∈ Ê^{+}(L_{m}) be a point whose image under j is 0. Then there exists Q ∈ Ê(L_{m}) such that P = p^{n}Q. Since ω_{m}^{+} is the exact annihilator of Ê^{+}(L_{m}) in Ê(L_{m}) (cf. Proposition 4.11 of [IP]) we have

p^{n}(ω_{m}^{+}Q) = 0,

and as there are no non-zero p-power torsion points in Ê(L_{M}), we conclude that

ω_{m}^{+}Q = 0.

Hence Q itself belongs to Ê^{+}(L_{m}), which proves the lemma.

Recall from Section 3.1 that local Tate duality induces perfect pairings

⟨ , ⟩_{m,n} : H^{i}(L_{m}, T_{E,n}) × H^{2−i}(L_{m}, T_{E,n}) ⟶ ℚ_{p}/ℤ_{p}

and

⟨ , ⟩_{m} : H^{i}(L_{m}, T_{p}E) × H^{2−i}(L_{m}, E[p^{∞}]) ⟶ ℚ_{p}/ℤ_{p}.

Let us define, following Section 4.3 of [IP],

H_{fin ±}^{1}(L_{m}, T_{f,n}) := (Ê^{±}(L_{m}) ⊗ ℤ/p^{n}ℤ),
H_{fin}^{1}(L_{m}, T_{f,n}) := (Ê(L_{m}) ⊗ ℤ/p^{n}ℤ),
H_{±}^{1}(L_{m}, T_{f,n}) := (Ê^{±}(L_{m}) ⊗ ℤ/p^{n}ℤ)^{⊥},

where the orthogonal complement in the last definition is taken relative to the pairing ⟨ , ⟩_{m}. We’ll also write

H_{±}^{1}(L_{m}, T_{E,n}) := (Ê^{±}(L_{m}) ⊗ ℤ/p^{n}ℤ)^{⊥},

with the orthogonal complement taken under the pairing ⟨ , ⟩_{m,n}.

Lemma 3.8. H^{0}(L_{m}, E[p^{n}]) = H^{2}(L_{m}, E[p^{n}]) = 0.

Proof. This follows from Lemma 4.6 of [IP] and the non-degeneracy of the local Tate pairing.

Lemma 3.9. H_{±}^{1}(L_{m}, T_{f,n}) is a free ℤ/p^{n}ℤ[G_{m}]-module of rank 1.

Proof. Taking the L_{m}-cohomology of the exact sequences

0 → T_{p}E ⟶^{p^{n}} T_{p}E ⟶ T_{E,n} → 0, 0 → T_{E,n} ⟶ E[p^{∞}] ⟶^{p^{n}} E[p^{∞}] → 0

and using Lemma 3.8 yields the natural isomorphisms of ℤ/p^{n}ℤ[G_{m}]-modules

H^{1}(L_{m}, T_{E,n}) ≅ H^{1}(L_{m}, T_{p}E)/p^{n}H^{1}(L_{m}, T_{p}E)

and

H^{1}(L_{m}, T_{E,n}) ≅ H^{1}(L_{m}, E[p^{∞}])[p^{n}].

The pairing ⟨ , ⟩_{m,n} is naturally induced from the pairing ⟨ , ⟩_{m} under these identifications. Therefore, since Ê^{±}(L_{m}) ⊗ ℚ_{p}/ℤ_{p} is a p-divisible group we immediately obtain

H_{±}^{1}(L_{m}, T_{E,n}) ≅ (Ê^{±}(L_{m}) ⊗ ℤ/p^{n}ℤ)^{⊥}
≅ (Ê^{±}(L_{m}) ⊗ ℚ_{p}/ℤ_{p})^{⊥}/p^{n}(Ê^{±}(L_{m}) ⊗ ℚ_{p}/ℤ_{p})^{⊥}.

This yields the isomorphism

H_{±}^{1}(L_{m}, T_{E,n}) ≅ H_{±}^{1}(L_{m}, T_{p}E)/p^{n}H_{±}^{1}(L_{m}, T_{p}E).

Proposition 4.16 of [IP] implies that H_{±}^{1}(L_{m}, T_{p}E) is a free ℤ_{p}[G_{m}]-module of rank 1. Lemma 3.9 follows.

The following result, which is a consequence of Theorem 10.1 of [GIP], shows that the subgroup H_{fin}^{1}(K_{m,p}, T_{f,n}) depends only on the Galois representation T_{f,n}, unlike its counterpart for ℓ ≠ p in general, so that in particular it behaves well under congruences. As [GIP] is not yet available we will sketch here the main arguments of the proof. Let f_{1}, f_{2} ∈ S_{2}(V/Γ) and let us denote by T_{i} := T_{f_{i}} and T_{i,n} := T_{f_{i},n} for i = 1, 2 and some n ≥ 1.

Theorem 3.10. Suppose that the G_{ℚ_{p}}-representations T_{1} and T_{2} are congruent modulo p^{n}, i.e. we have an isomorphism

ι : T_{1,n} ≅ T_{2,n}

as ℤ/p^{n}ℤ[G_{ℚ_{p}}]-modules. We’ll further assume that Assumptions 3.5 hold for T_{1,n} (and consequently also for T_{2,n}) and that L is one of the local fields K_{m,𝔭} or K_{m,𝔭̄} for some m ≥ 0. Then ι induces a natural isomorphism

g_{L} : H_{fin}^{1}(L, T_{1,n}) ⟶ H_{fin}^{1}(L, T_{2,n}).

Proof. First we have natural isomorphisms and inclusions

H_{fin}^{1}(L, T_{i})/p^{n}H_{fin}^{1}(L, T_{i}) ≅ H_{fin}^{1}(L, T_{i,n}) ↪ H^{1}(L, T_{i,n}), for i = 1, 2

as a consequence of the assumptions and lemma 3.8. Therefore the conclusion of the theorem makes sense. Second, given the totally ramified extension L/ℚ_{p} we perform (see [Fa] and [Br]) the following construction. Let π be a uniformizer of L and let

E(u) = u^{e} + a_{1}u^{e−1} + ... + a_{e} ∈ ℤ_{p}[u]

be the minimal polynomial of π over ℤ_{p}. Let S denote the p-adic completion of the ℤ_{p}-algebra ℤ_{p}[u, u^{ie}/i!]_{i∈ℕ} ⊂ ℚ_{p}[u]. It has the following structure:

1. a continuous ℤ_{p}-linear Frobenius σ : S ⟶ S such that σ(u) = u^{p};

2. a natural continuous derivation d : S ⟶ Ω_{S/ℤ_{p}};

3. a decreasing filtration (Fil^{i}S)_{i∈ℕ}, where Fil^{i}S is the p-adic completion of ∑_{j≥i}(E(u)^{j}/j!)S. (One checks that Fil^{i}S is an ideal of S.)

Let us consider the Galois representations T_{1}, T_{2} as above, denote by D_{1}, D_{2} the strongly divisible lattices attached to T_{1}, T_{2} respectively as in [FL] and set M_{i} := D_{i} ⊗_{ℤ_{p}} S, for i = 1, 2. Let finally M be any one of the S-modules M_{1}, M_{2}, M_{1}/p^{n}M_{1}, or M_{2}/p^{n}M_{2}. Then M is endowed with the following structure:

1. a one step filtration M_{0} ⊂ M;

2. σ-linear Frobenii φ : M → M and φ_{0} : M_{0} → M such that φ|_{M_{0}} = pφ_{0};

3. a connection (nilpotent modulo p) ∇ : M ⟶ M ⊗_{S} Ω_{S/ℤ_{p}} such that ∇ ∘ φ_{0} = (φ/p) ∘ ∇|_{M_{0}}.

Given M as above we define the double complex of ℤ_{p}-modules

\begin{aligned}& M_{0} & ⟶^{∇} & M ⊗_{S} Ω_{S/ℤ_{p}} \\ C^{•,•}(M) : & α ↓ & & ↓ β \\ & M & ⟶^{∇} & M ⊗_{S} Ω_{S/ℤ_{p}}, \end{aligned}

where α = φ_{0} − 1 and β = −(1 − φ/p). Let us write

H^{n}(M) := H^{n}(C^{•,•}(M)), for n ≥ 0.

Then under the conditions of the theorem one can prove (for the details see [GIP]) that H^{2}(M) = 0 and that there is a canonical and functorial isomorphism

H^{1}(M_{i}) ≅ H_{fin}^{1}(L, T_{i}) for i = 1, 2.

From the exact sequences

0 ⟶ T_{i} ⟶^{p^{n}} T_{i} ⟶ T_{i,n} ⟶ 0

for i = 1, 2 we deduce the isomorphisms:

H^{1}(M_{i}/p^{n}M_{i}) ≅ H_{fin}^{1}(L, T_{i})/p^{n}H_{fin}^{1}(L, T_{i}) ≅ H_{fin}^{1}(T_{i,n}).

The fact that T_{1,n} and T_{2,n} are isomorphic implies that the same is true for M_{1}/p^{n}M_{1} and M_{2}/p^{n}M_{2}. The result follows.

We can use Theorem 3.10 to meaningfully define the restricted local conditions at p for f. Namely, assume that the Assumptions 3.5 hold. In particular T_{f} is congruent to T_{p}E modulo p^{n} so we define H_{fin}^{1,±}(K_{m,p}, T_{f,n}) to be the image of Ê^{±}(K_{m,p})/p^{n}Ê^{±}(K_{m,p}) under the composition

Ê(K_{m,p})/p^{n}Ê(K_{m,p}) ≅ H_{fin}^{1}(K_{m,p}, T_{E,n}) ≅ H_{fin}^{1}(K_{m,p}, T_{f,n})

where the first isomorphism is the Kummer map and the second is provided by Theorem 3.10 via local duality. We also define the group H_{±}^{1}(K_{m,p}, T_{f,n}) to be the orthogonal complement of H_{fin}^{1,±}(K_{m,p}, T_{f,n}) under local duality. Clearly this group is the image of H_{±}^{1}(K_{m,p}, T_{E,n}) under the isomorphism H^{1}(K_{m,p}, T_{E,n}) ≅ H^{1}(K_{m,p}, T_{f,n}) induced by ι.

We recall that Λ_{n,m} := ℤ/p^{n}ℤ[G_{m}] = Λ_{m}/p^{n} denotes the group ring of G_{m} with mod p^{n} coefficients. From the previous discussion, we have

Corollary 3.11. The local cohomology groups H_{±}^{1}(K_{m,p}, T_{f,n}) are free Λ_{n,m}- modules of rank two.

Proof. Since p = 𝔭𝔭̄, we have

H_{±}^{1}(K_{m,p}, T_{f,n}) = H_{±}^{1}(K_{m,𝔭}, T_{f,n}) ⊕ H_{±}^{1}(K_{m,𝔭̄}, T_{f,n}).

But each of the summands on the right is a free Λ_{n,m}-module, by Lemma 3.9. The result follows.

3.3 Generalised Selmer groups

We are now in a position to define more general Selmer groups that will play a key role in our argument. We retain the notations and assumptions of earlier sections; in particular Assumption 3.5 of Section 3.2.

Let

Γ_{m} = Gal(K_{∞}/K_{m}), so that G_{m} = Γ/Γ_{m}.

If s belongs to H^{1}(K_{m}, T_{f,n}) and ℓ is a rational prime we denote by s_{ℓ} := res_{ℓ}(s) the image of s in H^{1}(K_{m,ℓ}, T_{f,n}) under restriction.

Definition 3.12. Let m and n be non-negative integers. The unrestricted Selmer group attached to f, n, and K_{m}, denoted Sel_{□}(K_{m}, T_{f,n}), is the group of classes s ∈ H^{1}(K_{m}, T_{f,n}) satisfying

s_{ℓ} ∈ H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) for all ℓ ≠ p.

The subgroups

Sel_{0}(K_{m}, T_{f,n}) ⊂ Sel_{±}(K_{m}, T_{f,n}) ⊂ Sel(K_{m}, T_{f,n}) ⊂ Sel_{□}(K_{m}, T_{f,n}) are defined by the additional conditions:

1. s ∈ Sel_{0}(K_{m}, T_{f,n}) if s_{p} = 0;

2. s ∈ Sel_{±}(K_{m}, T_{f,n}) if s_{p} ∈ H_{fin ±}^{1}(K_{m,p}, T_{f,n});

3. s ∈ Sel(K_{m}, T_{f,n}) if s_{p} ∈ H_{fin}^{1}(K_{m,p}, T_{f,n}).

We also define

Sel_{♯}(K_{∞}, T_{f,n}) := lim_{→,m} Sel_{♯}(K_{m}, T_{f,n}),
Sel_{♯}(K_{∞}, T_{f,∞}) := lim_{→,n} Sel_{♯}(K_{∞}, T_{f,n}),

where the transition maps for the inductive limit are restrictions and the inclusions T_{f,n} ↪ T_{f,n′}, and ♯ is either 0, ±, or □.

Let S be a square-free integer prime to N. Definition 3.13. The generalized Selmer group Sel_{S,♯}(K_{m}, T_{f,n}) is the set of classes in Sel_{♯}(K_{m}, T_{f,n}) satisfying

s_{ℓ} = 0 for all ℓ|S.

Definition 3.14. Let ♯ = 0, fin, ±, or □. The dual Selmer group attached to f, n, K_{m}, S and ♯ is defined to be the subgroup H_{S,♯}^{1}(K_{m}, T_{f,n}) of classes κ ∈ H^{1}(K_{m}, T_{f,n}) satisfying

1. κ_{ℓ} ∈ H_{fin}^{1}(K_{m,ℓ}, T_{f,n}) for all rational primes ℓ not dividing pS;

2. κ_{p} belongs to H_{♯}^{1}(K_{m,p}, T_{f,n}) if ♯ ∈ {fin, ±}, and κ_{p} = 0 if ♯ = 0;

3. κ_{ℓ} is arbitrary if ℓ|S.

Note the sequence of inclusions

H_{S,0}^{1}(K_{m}, T_{f,n}) ⊂ H_{S,fin}^{1}(K_{m}, T_{f,n}) ⊂ H_{S,±}^{1}(K_{m}, T_{f,n}) ⊂ H_{S,□}^{1}(K_{m}, T_{f,n}).

As in [BD2], denote by

Ĥ^{1}(K_{∞}, T_{f,n}) = lim_{←,m} H^{1}(K_{m}, T_{f,n}),

where the transition maps are co-restrictions. Similar conventions are adopted in defining Ĥ^{1}(K_{∞,p}, T_{f,n}).

Both Sel_{S,♯}(K_{m}, T_{f,n}) and H_{S,♯}^{1}(K_{m}, T_{f,n}) are special cases of the general notion of an abstract Selmer group: a subgroup of the global cohomology group H^{1}(K_{m}, T_{f,n}) defined by local conditions which agree with the unramified classes, for all but finitely many primes of K_{m}. The following pairs are dual Selmer groups in the sense that the local conditions defining them are orthogonal to each other under the local Tate pairings:

Sel_{S,0}(K_{m}, T_{f,n}) and H_{S,□}^{1}(K_{m}, T_{f,n});
Sel_{S,±}(K_{m}, T_{f,n}) and H_{S,±}^{1}(K_{m}, T_{f,n});
Sel_{S}(K_{m}, T_{f,n}) and H_{S,fin}^{1}(K_{m}, T_{f,n});
Sel_{S,□}(K_{m}, T_{f,n}) and H_{S,0}^{1}(K_{m}, T_{f,n}).

Finally we define

Ĥ_{S,♯}^{1}(K_{∞}, T_{f,n}) := lim_{←,m} H_{S,♯}^{1}(K_{m}, T_{f,n}),

where the transition maps are co-restrictions.

3.4 Freeness results for Selmer groups

The Euler system of this section is constructed (just like the Euler system of [BD2]) from a system of Heegner points on a collection of Shimura curves indexed by certain admissible primes ℓ. The main new difficulty arising in the supersingular setting is that the classes manufactured directly from these Heegner points are not compatible under norms (co-restriction). To obtain a norm-compatible family of cohomology classes it is necessary to divide the classes obtained “directly” from Heegner points by certain products of p-power cyclotomic polynomials, a process which mirrors the division performed in the construction of the p-adic L-function in Section 2 following [Po1]. In order to show that this division can be performed, a number of results concerning the structure of generalized Selmer groups as modules over the group rings Λ_{n,m} are required.

Let ℓ be an n-admissible prime. Note (cf. the discussion preceding Theorem 4.1 of [BD2]) the canonical direct sum decomposition:

Ĥ^{1}(K_{ℓ,∞}, T) = Ĥ_{fin}^{1}(K_{ℓ,∞}, T) ⊕ Ĥ_{sing}^{1}(K_{ℓ,∞}, T).

Denote (as in [BD2]) by v_{ℓ} and ∂_{ℓ} the projections onto the first and second factors. We recall the following proposition from [BD2] that makes it possible to produce many n-admissible primes.

Proposition 3.15. Let s be any nonzero element of H^{1}(K, T_{f,1}). There exist infinitely many n-admissible primes ℓ relative to f such that ∂_{ℓ}(s) = 0 and v_{ℓ}(s) ≠ 0.

Proof. This is Theorem 3.2 of [BD2], whose proof relies on a careful application of the Chebotarev density theorem, and makes no use of the local properties of T_{f,1} at p, so that it applies equally well to the supersingular case.

Adopting the terminology of Definition 2.22 of [BD1], we make the following definition.

Definition 3.16. A square-free product S of n-admissible primes is said to be n-admissible if the natural map

Sel_{□}(K, T_{f,n}) ⟶ ⊕_{ℓ|S}H_{fin}^{1}(K_{ℓ}, T_{f,n})

is injective.

Note that, if S is n-admissible, then

Sel_{S,□}(K, T_{f,n}) = 0.  (10)

Recall that Λ_{n,m} = ℤ/p^{n}ℤ[G_{m}] = Λ/(ω_{m}, p^{n})Λ is the group ring at level m with ℤ/p^{n}ℤ coefficients. Let I be the augmentation ideal of Λ_{n,m} and denote by 𝔪 = ⟨p, I⟩ the maximal ideal of this local ring. We begin by noting the following “control theorems” for the Selmer groups that have been introduced:

Lemma 3.17. If S is an n-admissible set, then the natural maps induced by restriction and the inclusion T_{f,1} ⟶ T_{f,n}

Sel_{S,□}(K, T_{f,1}) ⟶ Sel_{S,□}(K_{m}, T_{f,n})[𝔪],
H_{S,0}^{1}(K, T_{f,1}) ⟶ H_{S,0}^{1}(K_{m}, T_{f,n})[𝔪]

are isomorphisms.

Proof. Consider the following commutative diagram with exact rows:

0 → Sel_{S,□}(K, T_{f,n}) → H^{1}(K, T_{f,n}) → Ω_{S}(K)
↓ ↓ ↓
0 → Sel_{S,□}(K_{m}, T_{f,n})^{G_{m}} → H^{1}(K_{m}, T_{f,n})^{G_{m}} → Ω_{S}(K_{m}),

(11)

where

Ω_{S}(K) := (⊕_{ℓ|S}H^{1}(K_{ℓ}, T_{f,n})) ⊕ (⊕_{ℓ∤pS}H_{sing}^{1}(K_{ℓ}, T_{f,n})) ,
Ω_{S}(K_{m}) := (⊕_{ℓ|S}H^{1}(K_{m,ℓ}, T_{f,n})) ⊕ (⊕_{ℓ∤pS}H_{sing}^{1}(K_{m,ℓ}, T_{f,n})) .

The inflation-restriction sequence for T_{f,n} implies that the middle vertical map in (11) is an isomorphism, because T_{f,n}^{G_{m}} = 0. The rightmost vertical map is injective, by Lemma 3.2 and the fact that primes in S split completely in K_{m}/K. It follows from the five-lemma that Sel_{S,□}(K, T_{f,n}) = Sel_{S,□}(K_{m}, T_{f,n})^{G_{m}}, and therefore

Sel_{S,□}(K_{m}, T_{f,n})[𝔪] = Sel_{S,□}(K_{m}, T_{f,n})[I][p] =
= Sel_{S,□}(K, T_{f,n})[p] = Sel_{S,□}(K, T_{f,1}),

where the last equality follows from the fact that H^{0}(K, T_{f,n−1}) = 0. The proof of the second assertion, in which □ is replaced by 0, uses in addition the injectivity of the map H^{1}(K_{p}, T_{f,n}) ⟶ H^{1}(K_{m,p}, T_{f,n})^{G_{m}} in the analysis of the diagram analogue to diagram 11 (which follows from the fact that T_{f,n}^{G_{Km,p}} = 0), but is otherwise the same.

Lemma 3.18. If S is an n-admissible set, then Sel_{S,♯}(K_{m}, T_{f,n}) = 0, for all m and ♯ = 0, ±, fin, or □.

Proof. It suffices to show that the finite Λ_{n,m}-module M := Sel_{S,□}(K_{m}, T_{f,n}) is trivial, since this Selmer group contains all the others. By Lemma 3.17 and (10),

M[𝔪] = Sel_{S,□}(K, T_{f,1}) = 0,

and hence M = 0.

The following proposition gives explicit formulae for the cardinality of the global cohomology groups H_{S,♯}^{1}(K_{m}, T_{f,n}).

Proposition 3.19. Let t := #S − 2, and let δ_{m} := [K_{m} : K] = p^{m}. For all n-admissible sets S, and for all m ≥ 0,

#H_{S,0}^{1}(K_{m}, T_{f,n}) = p^{ntδ_{m}},
#H_{S,□}^{1}(K_{m}, T_{f,n}) = #H_{S,0}^{1}(K_{m}, T_{f,n})#H^{1}(K_{m,p}, T_{f,n}).

Proof. A general theorem arising from the Poitou-Tate exact sequence in Galois cohomology (cf. for example Theorem 2.19 of [DDT]) relates the cardinalities of a Selmer group and its dual, expressing the ratio of these cardinalities as a product of simple local terms. In the present context, Theorem 2.19 of [DDT] gives:

\frac{#H_{S,0}^{1}(K_{m}, T_{f,n})}{#Sel_{S,□}(K_{m}, T_{f,n})} = ( ∏_{ℓ|S∞} \frac{#H^{1}(K_{m,ℓ}, T_{f,n})}{#H^{0}(K_{m,ℓ}, T_{f,n})}) ,

\frac{#H_{S,□}^{1}(K_{m}, T_{f,n})}{#Sel_{S,0}(K_{m}, T_{f,n})} = ( ∏_{ℓ|S∞} \frac{#H^{1}(K_{m,ℓ}, T_{f,n})}{#H^{0}(K_{m,ℓ}, T_{f,n})}) × #H^{1}(K_{m,p}, T_{f,n}).

By Lemma 3.18, the denominators occurring in the left-hand sides of these formulae are equal to 1. Furthermore, we already know from Lemma 3.4 that

\frac{#H^{1}(K_{m,ℓ}, T_{f,n})}{#H^{0}(K_{m,ℓ}, T_{f,n})} = \{ \begin{matrix}p^{nδ_{m}} & if v|S; \\ p^{−2nδ_{m}} & if v = ∞. \end{matrix}

The Proposition follows.

The usefulness of the concept of n-admissible set lies in the following two propositions concerning the groups H_{S,0}^{1}(K_{m}, T_{f,n}) and H_{S,±}^{1}(K_{m}, T_{f,n}). These propositions can be viewed as global analogues of Lemma 3.4 and Corollary 3.11.

Proposition 3.20. If S is an n-admissible set, then the group H_{S,0}^{1}(K_{m}, T_{f,n}) is free of rank t := #S − 2 over Λ_{n,m}.

Proof. Consider the module M := (H_{S,0}^{1}(K_{m}, T_{f,n}))^{∨}, where the superscript ∨ denotes the Pontryagin dual. We have

M/𝔪M = (H_{S,0}^{1}(K_{m}, T_{f,n})[𝔪])^{∨} = (H_{S,0}^{1}(K, T_{f,1}))^{∨} ≃ (ℤ/pℤ)^{t},

where the second equality follows from Lemma 3.17 and the third isomorphism from Proposition 3.19 with m = 0 and n = 1. Let ξ_{1}, …, ξ_{t} be a set of elements of M which map to a basis for M/𝔪M. By Nakayama’s lemma, these elements generate M as a Λ_{n,m}-module, and yield a surjective map Λ_{n,m}^{t} → M of Λ_{n,m}-modules. Proposition 3.19 implies that #M = #(Λ_{n,m}^{t}), and hence this map is an isomorphism. It follows that the module M is free of rank t, and therefore

H_{S,0}^{1}(K_{m}, T_{f,n}) ≃ (Λ_{n,m}^{∨})^{t}.

The local ring Λ_{m} := ℤ_{p}[G_{m}] = ℤ_{p}[t]/(t^{p^{m}} −1) is a local complete intersection in the sense of Definition 5.1 of [DDT]. Hence by Proposition 5.9 of [DDT], it is Gorenstein in the sense of Definition 5.8 in [DDT], i.e.,

Hom_{ℤ_{p}}(Λ_{m}, ℤ_{p}) ≃ Λ_{m}.

It follows that

Hom_{ℤ/p^{n}ℤ}(Λ_{n,m}, ℤ/p^{n}ℤ) ≃ Λ_{n,m}.

This proves Proposition 3.20.

Proposition 3.21. If S is an n-admissible set, then the group H_{S,±}^{1}(K_{m}, T_{f,n}) is free of rank #S over Λ_{n,m}.

Proof. The natural sequence

0 ⟶ H_{S,0}^{1}(K_{m}, T_{f,n}) ⟶ H_{S,□}^{1}(K_{m}, T_{f,n}) ⟶ H^{1}(K_{m,p}, T_{f,n}) ⟶ 0

is exact. This assertion follows from the definition of the objects involved, for all but the penultimate map, whose surjectivity is a consequence of the second assertion in Proposition 3.19. It follows that the sequence

0 ⟶ H_{S,0}^{1}(K_{m}, T_{f,n}) ⟶ H_{S,±}^{1}(K_{m}, T_{f,n}) ⟶ H_{±}^{1}(K_{m,p}, T_{f,n}) ⟶ 0

is exact. Proposition 3.21 now follows from Corollary 3.11 and Proposition 3.20.

4 Construction of the Euler system

We maintain the notations of the previous sections, and fix an n-admissible prime ℓ for f. Let X := X_{N^{+},N^{−}ℓ} be the Shimura curve introduced in Section 5.1 of [BD2], and let P_{m} ∈ X(K̃_{m}) be the Heegner point of conductor p^{m} defined in Section 6 of [BD2], where the integers which are denoted M^{+} and M^{−} in that section are set to be equal to N^{+} and N^{−}ℓ respectively. Note that, unlike the setting that is considered in Section 6 of [BD2], the integer M^{+} is now assumed to be prime to p. The behaviour of Heegner points under norms (cf. for example Proposition 3.10 of [Da]) implies that the Heegner points P_{m} satisfy the following compatibilities (expressed as equalities of divisors on X) for all m ≥ 1:

Tr_{m}^{m+1}(P_{m+1}) = T_{p}P_{m} − P_{m−1},

where Tr_{m}^{m+1} denotes the Galois trace from the m+1-st layer to the m-th layer (i.e., from K̃_{m+2} to K̃_{m+1}, or from K_{m+1} to K_{m}), and T_{p} is the p-th Hecke operator. Let g be an eigenform of weight 2 on X such that T_{g,n} ≅ T_{f,n} = T as Galois modules. Such an eigenform exists by condition 4 in the definition of n-admissible primes: cf. Proposition 3.12 of [BD2].

In order to replace the points P_{m} on X by degree zero divisors, we choose a fixed auxiliary prime q which does not divide N, ℓ, or p, and let

P̃_{m} := (T_{q} − (q + 1))P_{m} ∈ Div^{0}(X).

Denote by the same symbol the image of this divisor in the Jacobian Jac(X) of X.

Let κ̃(ℓ)_{m} denote the image of P̃_{m} in H^{1}(K̃_{m}, Jac(X)[p^{n}]) under the global Kummer map. Write κ(ℓ)_{m} for the image of κ̃(ℓ)_{m} under the composition

H^{1}(K̃_{m}, Jac(X)[p^{n}]) ⟶ H^{1}(K̃_{m}, T_{g,n}) ⟶ H^{1}(K_{m}, T_{g,n}) ≅ H^{1}(K_{m}, T).

The first map is induced by “projection onto the g-isotypical component” Jac(X)[p^{n}] ⟶ T_{g,n}, the second is co-restriction and the third is induced by the isomorphism T_{g,n} ≅ T.

Proposition 4.1. The element κ(ℓ)_{m} belongs to H_{ℓ,fin}^{1}(K_{m}, T).

Proof. Everything follows from Section 7 and the beginning of Section 8 of [BD2] except the behaviour of the class under localization at primes above p. Let 𝔭 be a prime of K_{m} above p, then by the properties of the Kummer map and of co-restriction we have κ̃(ℓ)_{m,𝔭} ∈ H_{fin}^{1}(K_{m,𝔭}, T_{g,n}). Now apply Theorem 3.10 and deduce that κ̃(ℓ)_{m,𝔭} ∈ H_{fin}^{1}(K_{m,𝔭}, T). Hence κ(ℓ)_{m,p} belongs to H_{fin}^{1}(K_{m,p}, T).

The classes κ(ℓ)_{m} satisfy the compatibility relations under the trace maps

Tr_{m}^{m+1}(κ(ℓ)_{m+1}) = −κ(ℓ)_{m−1}.

Therefore we have

Lemma 4.2. Let ϵ denote the sign of (−1)^{m}. Then ω_{m}^{ϵ}κ(ℓ)_{m} = 0.

Proof. Let ξ_{k} denote the p^{k}-th cyclotomic polynomial in T + 1 as in Section 2, and suppose without loss of generality that m is even. Then

ω_{m}^{+}κ(ℓ)_{m} = ω_{m−2}^{+}ξ_{m}κ(ℓ)_{m} = ω_{m−2}^{+}Tr_{m−1}^{m}(κ(ℓ)_{m}) = −ω_{m−2}^{+}κ(ℓ)_{m−2}.

The result now follows by induction, using the fact that T κ(ℓ)_{0} = 0. The proof when m is odd is identical.

Let S be a square-free product of primes which is n-admissible in the sense of Definition 3.16. We can view the class κ(ℓ) as an element of the larger Λ_{n,m}-module H_{S,±}^{1}(K_{m}, T_{f,n}). It is useful to do so because of the following proposition:

Proposition 4.3. There exists a unique class

η(ℓ)_{m}^{ϵ} ∈ H_{S,ϵ}^{1}(K_{m}, T)/ω_{m}^{ϵ}H_{S,ϵ}^{1}(K_{m}, T)

such that

ω̃_{m}^{−ϵ}η(ℓ)_{m}^{ϵ} = κ(ℓ)_{m}.

Proof. This follows from the fact that H_{S,±}^{1}(K_{m}, T) is a free Λ_{n,m}-module by Proposition 3.21, using Lemma 4.2 and Lemma 2.7.

Now define the global cohomology classes indexed by the n-admissible primes ℓ:

κ(ℓ)_{m}^{+} := (−1)^{\frac{m}{2}} η(ℓ)_{m} ∈ H_{S,+}^{1}(K_{m}, T)/ω_{m}^{+} if m is even;  (12)
κ(ℓ)_{m}^{−} := (−1)^{\frac{m+1}{2}} η(ℓ)_{m} ∈ H_{S,−}^{1}(K_{m}, T)/ω_{m}^{−} if m is odd.

An argument identical to the one used in the proof of Lemma 2.9 shows that the sequences {κ(ℓ)_{m}^{+}}_{m even} and {κ(ℓ)_{m}^{−}}_{m odd} are compatible under corestriction, so that we can write

κ(ℓ)^{±} := lim_{←} κ(ℓ)_{m}^{±}.  (13)

This element belongs to

lim_{←} H_{S,±}^{1}(K_{m}, T)/ω_{m}^{±} ≅ lim_{←}(H_{S,±}^{1}(K_{m}, T) ⊗ Λ/(ω_{m}^{±}, p^{n})) = Ĥ_{S,±}^{1}(K_{∞}, T).

Let ℓ be an n-admissible prime dividing S. Note that both Ĥ_{fin}^{1}(K_{∞,ℓ}, T) and Ĥ_{sing}^{1}(K_{∞,ℓ}, T) are isomorphic to Λ/p^{n} by Lemma 3.4.

As in [BD2], the classes κ(ℓ)^{±} satisfy two key reciprocity laws relating them to the p-adic L-functions L_{f}^{±} defined in Section 2. The first reciprocity law concerns the properties of the class κ(ℓ)^{±} at the prime ℓ.

Proposition 4.4. The class κ(ℓ)^{±} satisfies:

v_{ℓ}(κ(ℓ)^{±}) = 0, ∂_{ℓ}(κ(ℓ)^{±}) ≡ L_{f}^{±} (mod p^{n}),

where the equality holds in Λ/p^{n}, up to multiplication by elements of ℤ_{p}^{×} and G_{∞}.

Proof. Let us fix m ≥ 0 and consider the following commutative diagram

H_{S,±}^{1}(K_{m}, T)/ω_{m}^{±} ⟶^{∂_{ℓ}} H^{1}(K_{m,ℓ}, T)/ω_{m}^{±} = Λ/(ω_{m}^{±}, p^{n})
↑ ↑ ↑
H_{S,±}^{1}(K_{m}, T) ⟶^{∂_{ℓ}} H^{1}(K_{m,ℓ}, T) = Λ/(ω_{m}, p^{n})
∪ || ||
H_{S,fin}^{1}(K_{m}, T) ⟶^{∂_{ℓ}} H^{1}(K_{m,ℓ}, T) = Λ/(ω_{m}, p^{n})

The proof of the first explicit reciprocity law given in Section 8 of [BD2] adapts without change to the classes κ(ℓ)_{m} considered here and yields

∂_{ℓ}(κ(ℓ)_{m}) = L_{m}

up to units in Λ_{n,m}. Moreover as

κ(ℓ)_{m} = (−1)^{[ \frac{m+1}{2} ]}ω̃_{m}^{∓}κ(ℓ)_{m}^{±},

we have

(−1)^{[ \frac{m+1}{2} ]}ω̃_{m}^{∓}∂_{ℓ}(κ(ℓ)_{m}^{±}) = L_{m} = (−1)^{[ \frac{m+1}{2} ]}ω̃_{m}^{∓}L_{m}^{±}

in ω̃_{m}^{∓}Λ/(ω_{m}, p^{n}). Using the isomorphism of Lemma 2.7 we conclude that

∂_{ℓ}(κ(ℓ)_{m}^{±}) = L_{m}^{±}

in Λ/(ω_{m}^{±}, p^{n}) up to units in this ring.

Let ℓ_{1} and ℓ_{2} be distinct n-admissible primes relative to f, such that p^{n} divides ℓ_{1} + 1 − ϵ_{1}a_{ℓ_{1}}(f) and ℓ_{2} + 1 − ϵ_{2}a_{ℓ_{2}}(f), for ϵ_{1} and ϵ_{2} equal to ±1. It is further assumed that the pair (ℓ_{1}, ℓ_{2}) is rigid in the sense of Section 3.3 of [BD2].

The second reciprocity law describes the localization of κ(ℓ_{1}) at ℓ_{2}. Note that this localization belongs to the finite part of the local cohomology group at ℓ_{2}.

Let B′ be the definite quaternion algebra of discriminant Disc(B)ℓ_{1}ℓ_{2}, let R′ be an Eichler ℤ[1/p]-order of level N^{+} in B′ and let Γ′ := (R′)^{×}/ℤ[1/p]^{×}. The theory of congruences between modular forms yields the following proposition:

Proposition 4.5. There exists an eigenform g ∈ S_{2}(T /Γ′, ℤ/p^{n}ℤ) such that the following equalities modulo p^{n} hold:

T_{q}g ≡ a_{q}(f)g (q |̸Nℓ_{1}ℓ_{2}), U_{q}g ≡ a_{q}(f)g (q|N),  (14)
U_{ℓ_{1}}g ≡ ϵ_{1}g, U_{ℓ_{2}}g ≡ ϵ_{2}g.

Furthermore (because of the assumption that the pair (ℓ_{1}, ℓ_{2}) is rigid) the form g can be lifted to an eigenform with coefficients in ℤ_{p} satisfying (14) above. This form is p-isolated.

Proof. The existence of the mod p^{n} eigenform g, which relies on the concepts and notations introduced in Sections 5 and 9 of [BD2], is proved in Theorem 9.3 of [BD2].

For any class κ ∈ Ĥ_{S,±}^{1}(K_{∞}, T), and any n-admissible prime ℓ which does not divide S, write v_{ℓ}(κ) for the natural image of κ in Ĥ_{fin}^{1}(K_{∞,ℓ}, T_{f,n}) under the restriction map at ℓ. Note again that the target module for v_{ℓ} is isomorphic to Λ/p^{n}Λ by Lemma 2.7 of [BD2]. With these notations we are ready to state the second explicit reciprocity law.

Proposition 4.6. The equality

v_{ℓ_{2}}(κ(ℓ_{1})^{±}) = L_{g}^{±}

holds in Ĥ_{fin}^{1}(K_{∞,ℓ_{2}}, T_{f,n}) ≃ Λ/p^{n}Λ, up to multiplication by elements of ℤ_{p}^{×} and G_{∞}.

Proof. This is essentially Theorem 4.2 of [BD2], whose proof, explained in Section 9 of that article, adapts to the setting where a_{p} = 0, the class κ(ℓ_{1}) is replaced by κ(ℓ_{1})^{±} and L_{g} is replaced by L_{g}^{±}.

We record the following consequence of Propositions 4.4 and 4.6:

Corollary 4.7. For all pairs of n-admissible primes (ℓ_{1}, ℓ_{2}) attached to f, the equality

v_{ℓ_{1}}(κ(ℓ_{2})^{±}) = v_{ℓ_{2}}(κ(ℓ_{1})^{±})

holds in Λ/p^{n}Λ, up to multiplication by elements of ℤ_{p}^{×} and G_{∞}.

5 Proof of the main result

Following Section 2.1 of [BD2], we make the following assumptions on the mod p Galois representation attached to f which correspond to some of the hypotheses made in Assumption 1.6 on E.

Assumption 5.1. The Galois representation attached to T_{f,1} has image isomorphic to GL_{2}(𝔽_{p}). Furthermore, for all ℓ dividing N exactly, the Galois representation T_{f,1} has a unique Gal(ℚ̄_{ℓ}/ℚ_{ℓ})-stable one-dimensional subspace.

Thanks to the reciprocity laws given in proposition 4.4 and 4.6 of the previous section, the classes κ(ℓ)^{±} ∈ Ĥ_{{ℓ,±}}^{1}(K_{∞}, T_{f,n}) indexed by the nadmissible primes attached to f enjoy exactly the same properties as the classes κ(ℓ) used in [BD2] in the study of the main conjecture in the ordinary case. They will be used to show:

Theorem 5.2. Let f be an eigenform in S_{2}(V/Γ) with coefficients in ℤ_{p} which is p-isolated and satisfies assumption 5.1 above. Then the characteristic power series of (Sel_{±}(K_{∞}, T_{f,∞}))^{∨} divides the p-adic L-function L_{p}^{±}(f, K).

The proof (as well as the statement!) of this theorem is identical to that of Theorem 4.4. of [BD2], after replacing Sel(K_{∞}, T_{f,∞}) by Sel_{±}(K_{∞}, T_{f,∞}) and κ(ℓ) by κ(ℓ)^{±}. Before launching into this proof, let us first make the following general comments.

1. Unlike the approach that is followed in [Ka], where p-Selmer groups are bounded via global cohomology classes whose local behaviour at the prime p is related to p-adic L-functions, our approach adapts to the supersingular setting the ideas of [BD2], where the p-Selmer group is controlled using global classes whose local behaviour at primes ℓ ≠ p is related to p-adic L-series. The main new difficulty in the supersingular case lies in the construction of the classes κ(ℓ)^{±} satisfying the same relation to the Pollack-style p-adic L-functions L_{f}^{±} and L_{g}^{±} as the classes κ(ℓ) did with L_{f} and L_{g}. With these classes in hand, the argument of [BD2] never involves the local behaviour of the Galois representations T_{f,n} and T_{f,n} at p, but only at n-admissible primes ℓ ≠ p which split completely in K_{∞}/K. This is why the Euler system argument in the proof of Theorem 4.4 of [BD2] extends to the supersingular setting without raising new difficulties or requiring substantial modifications.

2. Our approach to Theorem 5.2 is to prove it by induction, reducing the statement about f to an identical one about g for an appropriately chosen modular form g which is congruent to f modulo p^{n}. It is for this reason that a more general main conjecture (applying to all modular eigenforms on definite quaternion algebras with ℤ_{p}-coefficients, and not just those associated to elliptic curves) is needed even if one is only interested in establishing Theorem 1.4 of the introduction.

Proof of Theorem 5.2: For the convenience of the reader, we recall here the main lines of the argument, with an emphasis on the aspects that are specific to the supersingular setting. Note however that we follow the strategy of the proof of Theorem 4.4 of [BD2] very closely, and that the modifications that need to be made to this proof are comparatively minor.

Proposition 3.1 of [BD2], implies that it is enough to show that

φ(L_{f}^{±})^{2} belongs to Fitt_{O}((Sel_{±}(K_{∞}, T_{f,n}))^{∨} ⊗_{φ} O),  (15)

for all homomorphisms φ of Λ into a discrete valuation ring O. Fix O, φ, and n, write π for a uniformiser of O, and let e := ord_{π}(p) be absolute ramification degree of O. Write

t_{f} := ord_{π}(φ(L_{f}^{±})).

Assume without loss of generality that

1. t_{f} < ∞. (Otherwise, φ(L_{f}^{±}) = 0 and (16) is trivially verified.)

2. The group (Sel_{±}(K_{∞}, T_{f,n}))^{∨} ⊗ O is non-trivial. (Otherwise, its Fitting ideal is equal to O and (16) is trivially verified.)

We propose to show that

φ(L_{f}^{±})^{2} belongs to Fitt_{O}((Sel_{±}(K_{∞}, T_{f,n}))^{∨} ⊗_{φ} O)  (16)

by induction on t_{f}. We begin by using the classes κ(ℓ)^{±} to construct global cohomology classes that will be used to bound Sel_{±}(K_{∞}, T_{f,n}). Let ℓ be any (n + t_{f})-admissible prime, and let S be a square-free product of (n + t_{f})-admissible primes with ℓ|S. Let

κ(ℓ)^{±} ∈ Ĥ_{{ℓ},±}^{1}(K_{∞}, T_{f,n+t_{f}}) ⊂ Ĥ_{S,±}^{1}(K_{∞}, T_{f,n+t_{f}})

be the cohomology class attached to ℓ in (13), and denote by κ_{φ}(ℓ)^{±} the natural image of this class in

M := Ĥ_{S}^{1}(K_{∞}, T_{f,n+t_{f}}) ⊗_{φ} O.

Note that this module is free over O/p^{(n+t_{f})}, by Proposition 3.21. By Proposition 4.4,

ord_{π}(κ_{φ}(ℓ)^{±}) ≤ ord_{π}(∂_{ℓ}κ_{φ}(ℓ)^{±}) = ord_{π}(φ(L_{f}^{±})) = t_{f},

so that t := ord_{π}(κ_{φ}(ℓ)^{±}) ≤ t_{f}. Choose an element κ̃_{φ}(ℓ)^{±} ∈ M satisfying

π^{t}κ̃_{φ}(ℓ)^{±} = κ_{φ}(ℓ)^{±}.  (17)

Note that κ̃_{φ}(ℓ)^{±} is well defined modulo the π^{t}-torsion subgroup of M, which is contained in the kernel of the natural homomorphism

Ĥ_{S}^{1}(K_{∞}, T_{f,n+t_{f}}) ⊗_{φ} O ⟶ Ĥ_{S}^{1}(K_{∞}, T_{f,n}) ⊗_{φ} O.

Let κ′_{φ}(ℓ)^{±} denote the image of κ̃_{φ}(ℓ)^{±} in Ĥ_{S}^{1}(K_{∞}, T_{f,n}) ⊗ O. Note that this class does not depend on the choice of κ̃_{φ}(ℓ)^{±} satisfying (17). The key properties of the class κ′_{φ}(ℓ)^{±} are summarized in the following two Lemmas. 5.3 and 5.4 below.

Lemma 5.3. The class κ′_{φ}(ℓ)^{±} belongs to Ĥ_{ℓ,±}^{1}(K_{∞}, T_{f,n}) ⊗_{φ} O, and

1. ord_{π}(κ′_{φ}(ℓ)^{±}) = 0.

2. v_{ℓ}(κ′_{φ}(ℓ)^{±}) = 0, and ord_{π}(∂_{ℓ}κ′_{φ}(ℓ)^{±}) = t_{f} − t.

Proof. The fact that κ′_{φ}(ℓ)^{±} belongs to Ĥ_{ℓ,±}^{1}(K_{∞}, T_{f,n}) ⊗_{φ} O follows from the fact that κ(ℓ)^{±} belongs to Ĥ_{ℓ,±}^{1}(K_{∞}, T_{f,n+t_{f}}). Property 1 follows from the construction of κ′_{φ}(ℓ)^{±}, while property 2 is a direct consequence of Proposition 4.4.

Lemma 5.4. The residue ∂_{ℓ}(κ′_{φ}(ℓ)^{±}) belongs to the kernel of the natural homomorphism

η_{ℓ} : Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) ⊗_{φ} O ⟶ (Sel_{±}(K_{∞}, T_{f,n}))^{∨} ⊗_{φ} O.

Proof. The proof is the same as that of Lemma 4.6 of [BD2].

We now turn to the proof of (16) when t_{f} = 0—the basis for the induction argument.

Proposition 5.5. If t_{f} = 0, (i.e., L_{f}^{±} is a unit) then (Sel_{±}(K_{∞}, T_{f,n}))^{∨} is trivial.

Proof. The proof is the same as that of Proposition 4.7 of [BD2], which makes no use of the hypothesis that p is ordinary.

Turning now to the general case of equation (16), let Π be the set of rational primes ℓ satisfying the following conditions:

1. ℓ is (n + t_{f})-admissible.

2. The quantity t = ord_{π}(κ_{φ}(ℓ)^{±}) is minimal, among all primes satisfying condition 1.

Proposition 3.15 implies that Π is non-empty. Let t be the common value of ord_{π}(κ_{φ}(ℓ)^{±}) for all ℓ ∈ Π. Note that t ≤ t_{f} by definition.

Lemma 5.6. The integer t is strictly less than t_{f}.

Proof. See Proposition 4.8 of [BD2].

Before stating the next lemma, we need to recall the notion of rigid pairs of n-admissible primes that is defined in Section 3.3 of [BD2]. This notion relies on the Selmer group Sel_{S}(ℚ, W_{f}) attached to the 3-dimensional mod p representation W_{f} := ad_{0}(T_{f,1}) consisting of trace 0 endomorphisms of T_{f,1}, and to a square-free product S of 1-admissible primes. The definition of this Selmer group is the same as in Definition 3.5 of [BD2], except that, since W_{f} is not ordinary at p, but is crystalline, the group H_{fin}^{1}(ℚ_{p}, W_{f}) is defined to be the set of cohomology classes that are crystalline at p. With this change, it is still true that f is p-isolated precisely when Sel_{1}(ℚ, W_{f}) = 0. (This is just Proposition 3.6. of [BD2] whose proof applies just as well to the case where f is non-ordinary at p.) Following Definition 3.9 of [BD2], we say that a pair (ℓ_{1}, ℓ_{2}) of n-admissible primes is a rigid pair if Sel_{ℓ_{1}ℓ_{2}}(ℚ, W_{f}) is trivial.

Lemma 5.7. There exist primes ℓ_{1}, ℓ_{2} ∈ Π such that (ℓ_{1}, ℓ_{2}) is a rigid pair.

Proof. See Lemma 4.9 of [BD2] whose proof adapts without change to the supersingular setting.

Let (ℓ_{1}, ℓ_{2}) be a rigid pair of (n + t_{f})-admissible primes in Π, whose existence is guaranteed by Lemma 5.7. By Proposition 4.6, note that t = t_{g} = ord_{π}(φ(L_{g})), where g is the p-isolated eigenform in S_{2}(T /Γ′) attached to f and (ℓ_{1}, ℓ_{2}) through proposition 4.5.

Recall the Selmer group

Sel_{ℓ_{1}ℓ_{2},±}(K_{∞}, T_{f,n}) ⊂ Sel_{±}(K_{∞}, T_{f,n})

consisting of classes which are locally trivial at the primes dividing ℓ_{1} and ℓ_{2}. By definition, there is a natural exact sequence of Λ-modules

0 ⟶ S_{ℓ_{1}ℓ_{2}}^{f} ⟶ (Sel_{±}(K_{∞}, T_{f,n}))^{∨} ⟶ (Sel_{ℓ_{1}ℓ_{2},±}(K_{∞}, T_{f,n}))^{∨} ⟶ 0, (18)

where S_{ℓ_{1}ℓ_{2}}^{f} denotes the kernel of the natural surjection of duals of Selmer groups. Note the natural surjection given by local Tate duality:

η_{f} : (Ĥ_{sing}^{1}(K_{∞,ℓ_{1}}, T_{f,n}) ⊕ Ĥ_{sing}^{1}(K_{∞,ℓ_{2}}, T_{f,n})) ⟶ S_{ℓ_{1}ℓ_{2}}^{f}

induced from the inclusion

(S_{ℓ_{1}ℓ_{2}}^{f})^{∨} ⊂ H_{fin}^{1}(K_{∞,ℓ_{1}}, T_{f,n}) ⊕ H_{fin}^{1}(K_{∞,ℓ_{2}}, T_{f,n}).

The domain of η_{f} is isomorphic to (Λ/p^{n}Λ)^{2}, by Lemma 2.7 of [BD2]. Let η_{f}^{φ} denote the map induced from η_{f} after tensoring by O via φ. The domain of η_{f}^{φ} is isomorphic to (O/p^{n}O)^{2}. By Lemma 5.4, the kernel of η_{f}^{φ} contains the vectors (∂_{ℓ_{1}}κ′_{φ}(ℓ_{1})^{±}, 0) and (0, ∂_{ℓ_{2}}κ′_{φ}(ℓ_{2})^{±}) in

(Ĥ_{sing}^{1}(K_{∞,ℓ_{1}}, T_{f,n}) ⊕ Ĥ_{sing}^{1}(K_{∞,ℓ_{2}}, T_{f,n})) ⊗_{φ} O ≃ (O/p^{n}O)^{2}.

By part 3 of Lemma 5.3,

t_{f} − t_{g} = ord_{π}(∂_{ℓ_{1}}κ′_{φ}(ℓ_{1})^{±}) = ord_{π}(∂_{ℓ_{2}}κ′_{φ}(ℓ_{2})^{±}).

Hence

π^{2(t_{f} −t_{g})} belongs to the Fitting ideal of S_{ℓ_{1}ℓ_{2}}^{f} ⊗_{φ} O.  (19)

One may repeat the same argument with the modular form g. Thus we have an exact sequence similar to (18) but involving g instead of f:

0 ⟶ S_{ℓ_{1}ℓ_{2}}^{g} ⟶ (Sel_{±}(K_{∞}, T_{g,n}))^{∨} ⟶ (Sel_{ℓ_{1}ℓ_{2},±}(K_{∞}, T_{g,n})))^{∨} ⟶ 0, (20)

as well as a surjection given by local Tate duality:

η_{g} : (Ĥ_{fin}^{1}(K_{∞,ℓ_{1}}, T_{f,n}) ⊕ Ĥ_{fin}^{1}(K_{∞,ℓ_{2}}, T_{f,n})) ⟶ S_{ℓ_{1}ℓ_{2}}^{g}.

By global reciprocity, the kernel of the map η_{g}^{φ} obtained from η_{g} after tensoring by O via φ contains the elements

(v_{ℓ_{1}}κ′_{φ}(ℓ_{1})^{±}, v_{ℓ_{2}}κ′_{φ}(ℓ_{1})^{±}) = (0, v_{ℓ_{2}}κ′_{φ}(ℓ_{1})^{±})

as well as (v_{ℓ_{1}}κ′_{φ}(ℓ_{2})^{±}, 0). But

ord_{π}(v_{ℓ_{2}}κ′_{φ}(ℓ_{1})^{±}) = ord_{π}(v_{ℓ_{1}}κ′_{φ}(ℓ_{2})^{±}) = t_{g} − t = 0.

It follows that the module S_{ℓ_{1}ℓ_{2}}^{g} ⊗_{φ} O is trivial, and the natural surjection

(Sel_{±}(K_{∞}, T_{g,n}))^{∨} ⊗_{φ} O ⟶ (Sel_{ℓ_{1}ℓ_{2},±}(K_{∞}, T_{g,n}))^{∨} ⊗_{φ} O is an isomorphism.

(21)

Recall that, by Lemma 5.6,

t_{g} < t_{f},

and that the eigenform g satisfies all the hypotheses of Theorem 5.2. (The fact that g is p-isolated follows from the fact that (ℓ_{1}, ℓ_{2}) is a rigid pair of admissible primes.) By the induction hypothesis,

φ(L_{g})^{2} belongs to the Fitting ideal of (Sel_{±}(K_{∞}, T_{g,n}))^{∨} ⊗_{φ} O.  (22)

The theory of Fitting ideals implies that

π^{2t_{f}} = π^{2(t_{f} −t_{g})}π^{2t_{g}}
∈ Fitt_{O}(S_{ℓ_{1}ℓ_{2}}^{f} ⊗ O) Fitt_{O}((Sel_{±}(K_{∞}, T_{g,n}))^{∨} ⊗ O), by (19) and (22)
= Fitt_{O}(S_{ℓ_{1}ℓ_{2}}^{f} ⊗ O) Fitt_{O}((Sel_{ℓ_{1}ℓ_{2},±}(K_{∞}, T_{g,n}))^{∨} ⊗ O), by (21).

But note that Sel_{ℓ_{1}ℓ_{2},±}(K_{∞}, T_{g,n}) = Sel_{ℓ_{1}ℓ_{2},±}(K_{∞}, T_{f,n}) by definition, in light of the fact that the Galois modules T_{f,n} and T_{g,n} are isomorphic, and that the local conditions used to define the associated Selmer groups are the same outside of ℓ_{1} and ℓ_{2}. It follows that

π^{2t_{f}} ∈ Fitt_{O}(S_{ℓ_{1}ℓ_{2}}^{f} ⊗ O) Fitt_{O}((Sel_{ℓ_{1}ℓ_{2},±}(K_{∞}, T_{f,n}))^{∨} ⊗ O)
⊂ Fitt_{O}((Sel_{±}(K_{∞}, T_{f,n}))^{∨} ⊗ O), by (18).

Hence φ(L_{f}^{±})^{2} belongs to the Fitting ideal of (Sel_{±}(K_{∞}, T_{f,n}))^{∨} ⊗_{φ} O, and (16) is therefore proved. Theorem 5.2 follows.

Note finally that Theorem 1.4 follows from Theorem 5.2 specialized to the case where f has integer Hecke eigenvalues, and hence corresponds to a modular elliptic curve E via the Eichler-Shimura construction combined with the Jacquet-Langlands correspondence.

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