Selmer Groups of Hecke Characters and Chow Groups of Self Products of CM Elliptic Curves ^{∗}

Jonathan Dee

January 4, 1999

Contents

1 Introduction 1

2 Notations and Conventions 3

3 Background 4 3.1 p-adic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Grössencharacters and L-functions . . . . . . . . . . . . . . . . . . . 6 3.3 The Galois representations . . . . . . . . . . . . . . . . . . . . . . . . 7 3.4 Iwasawa Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Bounding Selmer Groups 14 4.1 Kato’s Reciprocity Map . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 The Selmer group and Kato’s explicit reciprocity law . . . . . . . . . 17 4.3 Connection with L-values . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.5 Descent to ℚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Application to Chow Groups 26 5.1 Results modulo torsion . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Integral results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3 Remarks on Ordinary Primes . . . . . . . . . . . . . . . . . . . . . . 31

1 Introduction

Formulae relating special values of L-fuctions to orders of Selmer groups have been in existence for about a century. Dirichlet’s formula can be recast in this form, as can the Birch and Swinnerton-Dyer conjecture from the 1960’s. Conjectural generalisations of these formulae were made by Deligne, Beilinson, then Bloch and Kato, extending the framework to the case of a general mixed motive over a number field. They are beautiful in their own right, but recently novel applications have begun to become apparent. The formula predicted by Bloch and Kato in the case of an adjoint representation is very important in deformation theory, and for the symmetric square of an elliptic curve appears in Wiles’s work [Wiles]. In this paper we shall give application to the study of Chow groups, after proving various formulae for Selmer groups attached to elliptic crurves with complex multiplication.

∗Mathematics Subject Classification 11G40(14C25)

Let E be an elliptic curve with complex multiplication by K, a quadratic imaginary extension of ℚ, and suppose that E is defined over K. If L(E/K, 1) is nonzero then it was shown in [CW] that E(K) is finite, and later Rubin showed in [Rubin] that the whole Selmer group is finite with order as expected by the Birch and Swinnerton-Dyer conjecture (up to a controlled constant).

On the other hand, if I_{𝔣} denotes the group of fractional ideals of K prime to 𝔣, the conductor of E, then there is a certain homomorphism

ψ : I_{𝔣} ⟶ K^{×},

the Grössencharacter attached to E. Hecke attached an L-function L(ψ, s) to ψ and proved both its analytic continuation and functional equation. It is a result of Deuring that there is an equality of L-functions

L(ψ, s)L(\overline{ψ}, s) = L(E/K, s).  (1)

One may also consider L-functions attached to the homomorphisms ψ^{k} for positive integers k and ask whether analogues of the Birch and Swinnerton - Dyer conjecture hold. This takes us into the domain of very general conjectures of Beilinson, and Bloch and Kato (see [BK] and [FPR]).

Our main result on L-functions is the following.

Theorem 1.0.1 Let k be a positive integer and p > k a prime where E has good supersingular reduction. Then

#H_{f}^{1}(K, A_{k}) = (\frac{L(\overline{ψ}^{k}, k)L(ψ^{k}, k)}{\overline{Ω}^{k}Ω^{k}})_{p} ,

with the understanding that both sides may be simultaneously infinite.

Here H_{f}^{1}(K, A_{k}) is the Selmer group defined in Section 3.3 and Ω^{k} is certain period of E. See Section 2 for the notation (x)_{p} for x ∈ \overline{ℚ}_{p}. It is not obvious that the expression in brackets is a rational integer, but this fact follows from Damerell’s Theorem (see Corollary 7.18 in [Rubin2]).

We further show that both sides of the above equality are finite if k ≥ 3, and extend this to the case k = 2 when E is assumed to be defined over ℚ. In the final two sections we bound the Selmer group of the dual representation by using the functional equation, then deduce results over ℚ when E is assumed to be defined over ℚ.

The method we use is modelled on Rubin’s approach to the Birch and Swinnerton - Dyer conjecture. When Rubin deals with the case of supersingular primes he makes use of Wiles’s explicit reciprocity law. This does not apply to the Galois representations we are interested in, so we must use Kato’s generalisation of Wiles’s result. This and the two variable Main Conjecture proved by Rubin are the key ingredients in our proof.

It is essential for us to use some p-adic Hodge theory - the correct Selmer group at a supersingular prime cannot even be defined without some use of B_{cris} and B_{dR}, and Kato’s reciprocity law makes use of the theory in its statement. The analogous results to ours when E has good ordinary reduction at p was proved by Li Guo in [LiGuo]. In this case the local condition at p in the definition of the Selmer group may be given without resorting to p-adic Hodge theory, and the proof is consequently less technically involved.

After the first part of this work (up to and including Section 4) was completed it came to the attention of the author that some similar results have been obtained independently by Han (see [Han]). We subsequently realised that his results together with results of Nekovář [N2] could be used to provide information about Chow groups of arbitrary codimension for self products of CM elliptic curves. Our main result in this direction is the following. See Chapter 5 for more introductory material.

Theorem 1.0.2 Let E be as above, with good reduction at a prime p > 3, and let X be the d-fold self product of E. Let i be a strictly positive integer. Assume that for all 0 ≤ n ≤ i − 1

• p^{2} − 1 ∤ n + 1 + p(n − 1) if E has supersingular reduction at p,

• p − 1 ∤ 2n if E has ordinary reduction at p.

(e.g. if p > 2i + 1).

Then

Im(cl_{X}(CH^{i}(X)) ∩ F^{2}H^{2i}(X_{et}, ℚ_{p}(i)) = 0,

where X is the d-fold product of E. The same is true for E ⊗_{ℚ} K.

Furthermore, if p > 2d + 1 then in H^{2i}(\overline{X}, ℤ_{p}(i)) we have that Im(cl_{X})_{tors} is finite, and zero for all

p ∤ (2π)^{j} \frac{L(ψ^{2j}, −j)}{Ω^{2j}}.

Here the filtration is induced by the Hoschild-Serre spectral sequence, and

cl_{X} : CH^{i}(X) ⟶ H^{2i}(\overline{X}_{et}, ℤ_{p}(i)) ⟶ H^{2i}(\overline{X}_{et}, ℚ_{p}(i))

is the étale cycle class map.

I would like to thank Jan Nekovář for his encouragement, frequent help, and tireless patience, and John Coates for very helpful comments.

2 Notations and Conventions

We fix embeddings of \overline{ℚ} into both \overline{ℚ}_{p} and ℂ.

If p is a rational prime and x ∈ \overline{ℚ}_{p}, then write

(x)_{p} = p^{v_{p}(x)},

where v_{p} is the p-adic valuation of \overline{ℚ}_{p} normalised so that v_{p}(p) = 1, with the convention that (0)_{p} = ∞.

If K is a number field and v is a prime of K, denote by K_{v} the completion of K at v. If F is a local or global field let O_{F} denote the ring of integers of F. If F is any field we shall write \overline{F} for some algebraic closure of F. If K/F is a Galois extension, write G(K/F) for the Galois group. Write G_{F} = G(\overline{F} /F) whenever F is a perfect field.

We use the notation 𝔽_{q} for the finite field with q elements.

As usual any Galois cohomology groups will always be taken to mean continuous cohomology groups. We normalise our Hodge-Tate weights so that for example ℂ_{p}(1) has Hodge-Tate weight 1 (rather than −1).

We apologise for using both the arithmetic and geometric Frobenius. This is because the geometric Frobenius is related to the f operator on D_{cris}) (this being related, of course, to the Frobenius on the reduction of the variety mod p via the crystalline cohomology), while the arithmetic Frobenius sits more naturally with the Grössencharacter.

3 Background

3.1 p-adic Theory

We refer the reader to [Fon1] for the construction and properties of Fontaine’s rings of p-adic periods B_{cris} and B_{dR}, and to [FPR] for the basics of their application to local Galois cohomology.

Let K be a finite extension of ℚ_{p}, and let K_{0} be the maximal absolutely unramified subextension of ℚ_{p} in K with Frobenius σ (lifting x ↦ x^{p}).

If R is any topological ring then by an R-representation of G_{K} we mean an R-module V of finite type equipped with a continuous R-linear action of G_{K}. We shall sometimes refer to a ℚ_{p}-representation as a p-adic representation.

We first generalise Proposition 3.8 of [BK] to the case of representations with coefficients. Let F be a finite extension of ℚ_{p}. Suppose that V is an F -representation of G_{K}, and let T ⊂ V be an O_{F} -lattice in V, such that the action of G_{K} on V makes T into a O_{F} -representation of G_{K}. Write A = V/T, a discrete O_{F} -module with linear action of G_{K}. We have the following:

Proposition 3.1.1 Suppose that F is an unramified extension of K. Then the Tate pairing with coefficients

H^{1}(K, A) × H^{1}(K, T^{∗}(1)) ⟶ F/O_{F} ,  (2)

where T^{∗} = Hom_{O_{F}} (T, O_{F}), is perfect and induces an isomorphism

\frac{H^{1}(K, A)}{H_{e}^{1}(K, A)} ˜ ⟶Hom_{O_{F}} (H_{g}^{1}(K, T^{∗}(1)), F/O_{F}).  (3)

Proof: First note that

Hom_{O_{K}}(O_{F} , O_{K}) ≅ O_{F} .  (4)

Indeed, given x ∈ O_{F}, consider the function given by

y ↦ T r_{F/K}(xyc),

where c generates the inverse different of F/K. By definition of the different and trace this map is well defined and O_{K}-linear. To prove that all maps as above are of this form, note that trace gives a perfect K-linear pairing on F, so any function as above is of the form

y ↦ T r_{F/K}(zy)

for some z ∈ F. By the definition of the inverse different z = cx for some x ∈ O_{F}. Next let M be any O_{F} -module of finite rank. The adjunction formula tells us

Hom_{O_{F}} (M, Hom_{O_{K}}(O_{F} , O_{K})) ≅ Hom_{O_{K}}(M, O_{K}),  (5)

where, for instance, if

g : M ⟶ O_{K}

is O_{K}-linear, then we may define, for each m ∈ M

f(t) : s ↦ g(st),

t ↦ f(t) being the desired O_{F} -linear map.

Substituting T for M into (4) and (5), and noting that both equalities will be Galois equivariant if we let Galois act on F and K trivially, we find that

Hom_{O_{F}} (T, O_{F}) ≅ Hom_{O_{K}}(T, O_{K}).

Similarly

Hom_{O_{F}} (T, F/O_{F}) ≅ Hom_{O_{K}}(T, K/O_{K}).

The result now follows formally from the classical Tate duality pairing, and the result of Bloch and Kato. □

See [BK] for the definitions of the exponential and dual exponential maps for p-adic representations

exp_{V} : \frac{D_{dR}(V)}{D_{dR}^{0}(V)} ⟶ H_{e}^{1}(K, V),

exp_{V}^{∗} : \frac{H^{1}(K, V)}{H_{g}^{1}(K, V)} ⟶ D_{dR}^{0}(V).

They are natural in the variable V, so respect the action of F if V is an F - representation for F as above. One has the following formulae (see for example [N1]), whereall dimensions are taken over ℚ_{p}.

Lemma 3.1.2

dim H_{e}^{1}(K, V) = dim(D_{dR}(V)/D_{dR}^{0}(V)) + dim H^{0}(K, V) − dim D_{cris}(V)^{f=1},
dim H_{f}^{1}(K, V) = dim(D_{dR}(V)/D_{dR}^{0}(V)) + dim H^{0}(K, V),
dim H_{g}^{1}(K, V) = dim(D_{dR}(V)/D_{dR}^{0}(V)) + dim H^{0}(K, V) + dim D_{cris}(V^{∗}(1))^{f=1}.

Remark 3.1.3 Counting dimensions in the exact sequence of ℚ_{p}-vector spaces

0 ⟶ D_{cris}(V)^{f=1} ⟶ D_{cris}(V) ⟶ (1 − f)D_{cris}(V) ⟶ 0

shows that dim D_{cris}(V)^{f=1} = dim D_{cris}(V)/(1 − f)D_{cris}(V).

We shall need to understand the characteristic polynomial of the crystalline Frobenius acting on D_{cris}(V). Let X be a smooth projective variety defined over K and supppose X has a smooth projective model over O_{K}. If l is any prime write V_{l,n} = H_{et}^{n}(X\overline{K}, ℚ_{l}). Write H_{cris}^{n}(X) for the n^{th} rational crystalline cohomology group of the special fibre of a smooth projective model of X over O_{K}. It is a K_{0}-vector space with a σ-semilinear Frobenius operator f. Thus if a = [K_{0} : ℚ_{p}], then f^{a} acts σ-linearly. Since X has good reduction the G_{K}-representation V_{l} is unramified, so if F rob denotes a geometric Frobenius in G_{ℚ_{p}}, then there is a well defined action of F rob^{a} on V_{l,n}. Katz-Messing [KM] deduced the following from the Weil conjectures.

Fact 3.1.4 If l ≠ p

char(1 − T f^{a} | H_{cris}^{n}(X)) = char(1 − T F rob^{a} | V_{l,n}).

Combining this with the Crystalline Conjecture proved by Faltings (and Fontaine - Messing, Kato, Tsuji) that D_{cris}(V_{p}) is naturally isomorphic to H_{cris}(X), in a maner compatible with the action of Frobenius, we deduce

Lemma 3.1.5 We have

char(1 − T f^{a} | D_{cris}(V_{p})) = char(1 − T F rob^{a} | V_{l,n}).
□

Deligne’s proof of the Weil conjectures tells us that the roots of these characteristic polynomials are algebraic integers, all of whose conugates have absolute value equal to p^{−an/2}.

In fact we shall only use Lemma 3.1.5 when V_{l,n} is the rational Tate module of an elliptic curve with complex multiplication, and in this special case it can be deduced in an ad hoc way from the results of Fontaine in [Fon3].

3.2 Grössencharacters and L-functions

For background information on elliptic curves with complex multiplication see chapter 2 of [Sil2].

From now on let K be a quadratic imaginary extension of ℚ of discriminant d_{K}, and let E be an elliptic curve over K of conductor 𝔣 with complex multiplication by the maximal order in K. The field K then has class number 1. Fix an embedding of K into ℂ, and a minimal Weierstrass equation for E. This gives rise to a lattice L in ℂ and uniformisation

ℂ/L ≅ E(ℂ).  (6)

By the theory of complex multiplication we may choose Ω ∈ ℂ such that L = ΩO_{K}. The induced action of End_{K}(E) on the tangent space of E induces an identification of K with End_{K}(E) ⊗_{ℤ} ℚ which we suppress from the notation.

Attached to E is a Grössencharacter

ψ : I_{𝔣} ⟶ K^{×} ⊂ ℂ,

such that for every prime ideal 𝔭 of K, ψ(𝔭) is a generator of 𝔭 and ψ gives the action of the arithmetic Frobenius at 𝔭 on the Tate module of E at primes l with 𝔭 ∤ l.

Hecke proved analytic continuation and the functional equation for L-functions attached to Grössencharacters. If Λ denotes the extended L-function then

Λ(\overline{ψ}^{k}, s) = ϵ(\overline{ψ}, s)Λ(ψ^{k}, k + 1 − s),  (7)

where

ϵ(\overline{ψ}^{k}, s) = W (\overline{ψ}^{k})(| d_{K} | ·N_{K/ℚ}𝔣)^{1/2−s}.

Here d_{K} is the discriminant of K, and W (\overline{ψ}^{k}) is the root number which in our situation is ±1.

The connection between the L-functions of ψ^{k} and \overline{ψ}^{k} is

\overline{L(ψ^{k}, \overline{s})} = L(\overline{ψ}^{k}, s).  (8)

If E happens to be defined over ℚ then by considering conjugation of Frobenius elements by complex conjugation one shows that ψ(\overline{𝔭}) = \overline{ψ}(𝔭), Hence for s giving

L(ψ, s) = L(\overline{ψ}, s).  (9)

Combining (8) with (9) we deduce that if E is defined over ℚ then L(ψ^{k}, k) is real.

3.3 The Galois representations

From now on we suppose that E has good supersingular reduction at the prime 𝔭 of K with 𝔭 ∤ 6. It is known that this is equivalent to assuming that 𝔭 is inert: 𝔭 = pO_{K} for some rational prime p. For simplicity we identify p with 𝔭. It will be clear in any given context whether p means a rational prime or the prime in O_{K} which it generates. Recall that another equivalent condition for an elliptic curve to have supersingular reduction at a prime 𝔭 above p is that the Tate module of p-power torsion points on the reduced curve should vanish. Write O_{P} for the ring of integers in K_{p}.

We shall be interested in various representations related to the Tate module of E, T_{E}. Let T_{E} denote the ℤ_{p}-adic Tate module of E, let V_{E} = T_{E} ⊗_{ℤ_{p}} ℚ_{p}, and define V_{k} = V_{E}^{⊗k}, a free module over K_{p} of rank 1 with continuous action of G_{K}, the tensor product being taken over K_{p}. Similarly define T_{k} = T_{E}^{⊗k}, the tensor product being over O_{p}, and A_{k} = V_{k}/T_{k}.

We have fixed an embedding of \overline{ℚ} into \overline{ℚ}_{p}, and this gives an injection G_{ℚ_{p}} ↪ G_{ℚ}. In particular we may consider the restriction of the G_{K}-representations T_{k}, V_{k} and A_{k} to G_{K_{p}}. Since E has good reduction at p the G_{K_{p}}-representation V_{E} is crystalline. It follows from [Fon2] that V_{k} and V_{k}^{∗}(1) are also crystalline.

Let V be an arbitrary p-adic representation of G_{F}, where F is a number field. Suppose we are given a Galois equivariant ℤ_{p}-lattice T in V, and put A = V/T. Define the Selmer group for A over K to be

H_{f}^{1}(F, A) = ker (H^{1}(F, A) ⟶ ∏_{v prime of F} \frac{H^{1}(F_{v}, A)}{L_{v}} ) ,

where L_{v} = i_{v}(W_{v}),

i_{v} : H^{1}(F_{v}, V) ⟶ H^{1}(F_{v}, A),

and

W_{v} = \begin{cases}ker(H^{1}(F_{v}, V) ⟶ H^{1}(I_{v}, V)) & v ∤ p \\ H_{f}^{1}(F_{v}, V) & v = p \end{cases} ,

I_{v} being the inertia subgroup of G_{F_{v}}.

We shall also need the restricted Selmer group. This is given by

H_{f}^{1}(F, A)′ = ker (H^{1}(F, A) ⟶ ∏_{v∤p} \frac{H^{1}(F_{v}, A)}{L_{v}} ) .

The connection between these two Selmer groups may then be summarised by the exact sequence

0 ⟶ H_{f}^{1}(F, A) ⟶ H_{f}^{1}(F, A)′ ⟶ ⨁_{v|p} \frac{H^{1}(F_{p}, A)}{H_{f}^{1}(F_{p}, A)}.  (10)

In Section 5 we shall be interested in the Galois representations W_{k} = V_{2k}(1−k) and W_{k}^{∗}(1) for positive integers k. We now intend to calculate the dimensions of the local cohomology groups

H_{x}^{1}(K_{p}, V_{k}), H_{x}^{1}(K_{p}, V_{k}^{∗}(1))
H_{x}^{1}(K_{p}, W_{k}), H_{x}^{1}(K_{p}, W_{k}^{∗}(1))

for x = e, f or g. Since we shall have to prove results which are sometimes identical for V_{k} and W_{k} and sometimes slightly different we shall write U for either W_{k} or V_{k}, and similarly T_{U} and A_{U} for their canonical lattices and co-lattices. The first step is to understand D_{cris}(U) and D_{cris}(U^{∗}(1)) over K_{p}. Both K_{p}-modules are in fact K_{p} ⊗_{ℚ_{p}} K_{p}-modules, where the first copy of K_{p} is acting by functoriality (from the

complex multiplication of E), and the second copy is K_{p} = B_{cris}^{G_{Kp}}, which acts on (B_{cris} ⊗_{ℚ_{p}} U)^{G_{Kp}} in the natural way. Since the Frobenius f is linear with respect to the first K_{p}, anti-linear with respect to the second, and dim_{ℚ_{p}} D_{cris}(U) = 4, we deduce that D_{cris}(U) and D_{cris}(U^{∗}(1)) are free K_{p} ⊗_{ℚ_{p}} K_{p}-modules of rank 1. Since (ψ(p)) = (p) we have ψ(p) = up for some unit u in O_{K}. From section 3.2 we know that the geometric Frobenius class F rob_{p} in G_{K_{p}} acts on the rational l-adic Tate-module of E V_{E,l} by ψ^{−1} (for l ≠ p), so by Lemma 3.1.5 we find

char(1 − T f^{2} | D_{cris}(V_{k})) = (1 − (up)^{−k}T)^{2},  (11)

Now, if E is defined over ℚ then it is known that u = −1. We shall only need this fact when k = 2, so we make the following

Hypothesis 3.3.1 If k = 2 assume that E is defined over ℚ.

Thus f^{2} acts on D_{cris}(V) by (up)^{−k}. Using the Weil pairing we have

V_{k}^{∗}(1) ≅ V_{k}(1 − k),  (12)

so f^{2} acts on D_{cris}(V^{∗}(1)) by u^{k}p^{2−k}. It furthermore follows that f acts on D_{cris}(V_{1}) with respect to a suitable K_{p} ⊗ K_{p}-basis by

(\begin{matrix}0 & 1 \\ −1/p & 0 \end{matrix}) ,

and hence f acts on D_{cris}(W_{k}^{∗}(1)) by (−1)^{k}σ, a fact which will be needed in Section 5.

By (11) and the above determination of the structure of D_{cris}(U) as K_{p} ⊗ K_{p}- module equipped with the action of Frobenius we deduce from the above discussion:

Proposition 3.3.2

D_{cris}(V_{k})^{f=1} = D_{cris}(W_{k})^{f=1} = 0,
D_{cris}(V_{k}^{∗}(1))^{f=1} = 0 for k = 1 and k ≥ 3,
dim_{K_{p}} D_{cris}(W_{k}^{∗}(1))^{f=1} = 1.

We next examine H^{0}(K_{p}, U) and H^{0}(K_{p}, U^{∗}(1)). We shall prove that both groups are trivial for all k.

Lemma 3.3.3 H^{0}(K_{p}, U) = H^{0}(K_{p}, U^{∗}(1)) = 0 for all k ≥ 1.

Proof: The hypothesis of complex multiplication implies that the groups are either trivial or equal to the whole space. We are thus reduced to proving that G_{K_{p}} acts nontrivially on U and U^{∗}(1). Were the action trivial then the Hodge-Tate weights of the representation would both equal zero, which is not the case. □

Kato proves in section 2.2.3 of [Kato] that

D_{dR}^{0}(V_{k}) ≅ (Cotan(Ê) ⊗_{ℤ_{p}} ℚ_{p})^{⊗_{Kp}k}

canonically, where Cotan(Ê) is the cotangent space for the formal group of E/K at the prime p, a free O_{p}-module of rank 1. hence

dim_{K_{p}} D_{dR}^{0}(V) = 1.  (13)

Since V_{k} and V_{k}^{∗}(1) are de Rham representations of G_{K_{p}}

dim_{K_{p}} D_{dR}(V_{k}) = dim_{K_{p}} D_{dR}(V_{k}^{∗}(1)) = 2.  (14)

We could also see the above from the fact that V is de Rham with Hodge-Tate weights 0 and k, but we shall need the above isomorphism anyway in the construction of the explicit reciprocity map. The representation W_{k} is de Rham with Hodge-Tate weights −k and k, so both D_{dR}^{0}(W_{k}) and D_{dR}^{0}(W_{k}^{∗}(1)) also have K_{p}-dimension equal to 1.

We are now in a position to prove:

Proposition 3.3.4 In the following all dimensions are with respect to the K_{p}-vector space structures.

dim H^{1}(K_{p}, U) = dim H^{1}(K_{p}, U^{∗}(1)) = 2.

If k ≠ 2 and x = e, f, or g then

dim H_{x}^{1}(K_{p}, V_{k}) = dim H_{x}^{1}(K_{p}, V_{k}^{∗}(1)) = 2.

dim H_{x}^{1}(K_{p}, W_{k}) = \begin{cases}1 & x = e, or f, \\ 2 & x = g. \end{cases}

dim H_{x}^{1}(K_{p}, W_{k}^{∗}(1)) = \begin{cases}0 & x = e, \\ 1 & x = f, or g. \end{cases}

Proof: Tate’s calculation of the local Euler-Poincaré characteristic and local duality gives

dim H^{1}(K_{p}, U) = 2 + dim H^{0}(K_{p}, U) + dim H^{0}(K_{p}, U^{∗}(1)) = dim H^{1}(K_{p}, U^{∗}(1)),

and both H^{0}’s vanish by Lemma 3.3.3. the rest of the formulae follow immediately by subtituting (13), Proposition 3.3.2, and Lemma 3.3.3 into the formulae of Proposition 3.1.2. □

3.4 Iwasawa Theory

Let K_{∞} = K(A) be the extension of K given by the kernel of

ψ^{k} : G_{K} ⟶ O_{p}^{×},  (15)

(which is of course also the kernel of \overline{ψ}^{k}). Hence

G_{∞} = G(K_{∞}/K) ⊂ O_{p}^{×}

canonically. We know from the theory of complex multiplication that so long as p is prime to 6𝔣, ψ and \overline{ψ} are actually isomorphisms (see Corollary 5.17 of [Rubin2]). We may thus decompose

G_{∞} = Δ × Γ,

where Δ is a finite abelian group of order prime to p, and Γ is (noncanonically) isomorphic to ℤ_{p}^{2} (recall k > 0).

Let K_{0} denote the fixed field of Γ ⊂ G_{∞}, so G(K_{∞}/K_{0}) ≅ Γ, and more generally write K_{n} for the fixed field of Γ^{p^{n}}. It is known that there is a unique (totally ramified) prime p_{n} of O_{K_{n}} above p. Write K_{n,p} for the completion of K_{n} at that prime.

Write Λ = ℤ_{p}[[G_{∞}]] for the Iwasawa algebra corresponding to G_{∞}. Since #Δ is prime to p, ℤ_{p}[Δ] is semi-simple. Let e_{χ} be the idempotent in ℤ_{p}[δ] corresponding to the character χ of an irreducible ℤ_{p}-representation of Δ. Then

Λ = ℤ_{p}[Δ][[Γ]] = ⨁_{χ} Λ_{χ},

where χ runs over all characters of irreducible ℤ_{p} representations of Δ and

Λ_{χ} = e_{χ}Λ = R_{χ}[[Γ]],

R_{χ} being either ℤ_{p} or the ring of integers in the unique unramified quadratic imaginary extension of ℤ_{p} according as dim(χ) = 1 or 2. More generally, we have a decomposition for any Δ-module Y:

Y = ⨁ Y^{χ},  (16)

where χ runs over all irreducible ℤ_{p}-representations of Δ and Y = e_{χ}Y. Write A_{k} for the kernel of the homomorphism

ψ^{k} : Λ_{χ_{k}} ⟶ O_{p}

deduced from (15). Here χ_{k} is the restriction of ψ^{k} to Δ.

We shall restrict attention from now on to values of k such χ_{k} is ℤ_{p}-irreducible. This is not too drastic because of the following.

Lemma 3.4.1 χ_{k} is irreducible as a ℤ_{p}-representation of Δ if and only if p + 1 ∤ k.

Proof: This follows easily from the fact that ψ is surjective, E[p] ≅ O/p, and that Aut(E[p]) ≅ (O/p)^{×} is a cyclic group of order p^{2} − 1. □

The following two simple results will be needed later.

Lemma 3.4.2 Identify E[p] with O/p. Then χ_{k} = \overline{χ}χ if and only if p^{2}−1 | k−1−p.

Proof: Suppose χ_{k} = \overline{ψ}ψ. We know that ψ maps G_{K} onto Δ = (O/p)^{∗}. Now complex conjugation induces the unique nontrivial continuous automorphism of K_{p} which is the Frobenius for K_{p}/ℚ_{p}. Hence if x ∈ Δ, c(x) = x^{p}. By the surjectivity of χ_{1}, χ_{k} = χ_{1}\overline{χ_{1}} if and only if x^{k−1} = x^{p} for all x, if and only if k − 1 ≡ p mod p^{2} − 1.

The converse is clear by the same reasoning. □

Lemma 3.4.3 Suppose p > k and p ∤ 6. Then the homomorphism

ψ^{k} : Λ ⟶ O_{p}

is surjective.

Proof: This follows from the fact that ψ : G_{K} → O_{p}^{×} is surjective, as remarked at the beginning of this section. Hence if p ∤ k, then 1 + pO_{p} is contained in the image of our map, and if the torsion subgroup of O_{p} is not contained in the image, then χ_{k} maps G_{K} into 𝔽_{p}^{×}. We saw in the proof of Proposition 3.4.1 that this cannot occur if p > k − 1. □

Hypothesis 3.4.4 Here we collect our assumptions on the prime p. In view of Lemmas 3.4.1 and 3.4.2 we suppose from now on that p > k. Recall that we are already assuming that p ∤ 6 is a prime of good supersingular reduction for E.

Some Λ-modules

We shall use use the following standard Λ-modules for the Δ×O_{K}-extension K_{∞}/K. See [Rubin] for further explanation.

• A_{∞} (p-parts of ideal class groups).

It is known that A_{∞} is a finitely generated torsion Λ-module.

• U_{∞} (principal local units).

It shown in [Win] that U_{∞} is finitely generated and torsion free as a Λ-module, and that U_{∞}^{χ} is free of rank 2 over Λ_{χ}. (Note that in order to apply Wintenberger’s results we use Lemma 3.4.2 which ensures that χ is not the cyclotomic character).

• E_{∞} (closure of global units).

Note that since the local units at level n, U_{n} split as the product of principal units and roots of unity of order prime to p, the image E_{n} of the global units in U_{n} may be considered equally as a subgroup or quotient of the global units.

• X_{∞}.

This Λ-module is finitely generated of rank 1, and has no nonzero pseudo-null submodules (see [Green]).

We shall also need the Iwasawa module of elliptic units. For more details on the construction and properties of these units see [Rubin2].

Firstly, if x is the coordinate function on E coming from the Weierstrass model fixed in Section 3.2, define for each ideal 𝔞 of K prime to 6 (𝔞 will always be such an ideal in this section)

Θ_{E,𝔞} = α^{−12}Δ(E)^{N𝔞−1} ∏_{P ∈E[𝔞]−O}(x − x(P))^{−6},

where (α) = 𝔞.

Θ_{E,𝔞} is independent of α since all units in O_{K} are killed by raising to the twelfth power. In fact in general it is independent of the choice of the Weierstrass model and defined over any field of definition of E.

Fix a set B of ideals of K prime to 𝔞𝔣 such that the Artin map induces a bijection of B with G(K(𝔣)/K). Let P_{𝔣} generate E[𝔣] as O_{K}-module, and define

Λ_{E,𝔞}(P) = ∏_{𝔟∈B} Θ_{E,𝔞}(ψ(𝔟)P_{𝔣} + P).

This formula gives a rational function, and one can check that it is in fact defined over K. It is independent of the choice of B, (but not necessarily of P_{𝔣}), and if P_{𝔟} ∉ E[𝔣] (where 𝔟 is as above), then Λ_{E,𝔞}(P_{𝔟}) is a global unit in K(E[𝔟]).

Recall that

ξ : ℂ/L ≅ E(ℂ)

is our fixed uniformisation of E(ℂ). It is compatible with the group structure. Since ψ(p^{n}) generates the ideal (p^{n}),

P_{n} = ξ(ψ(p^{n})^{−1}Ω)  (17)

will have exact order p^{n}. One may show that η_{𝔞,n} = Λ_{E,𝔞}(P_{n}) is a global unit in K_{n}, and moreover that for fixed 𝔞 the η_{𝔞,n} are norm compatible in the tower K_{∞}/K. Let C′_{n} denote the group of units in K_{n} generated as ℤ[G(K_{n}/K)]-module by the roots of unity in K_{n} and the η_{𝔞,n}, where 𝔞 runs over the ideals of K prime to 6p𝔣. Finally define C_{n} to be the closure of the image of C′_{n} in U_{n}, and

C_{∞} = lim_{⟵} C_{n}.

We have the following result giving the structure of C_{∞} as a Λ_{χ}-module. First we need a preliminary result.

Lemma 3.4.5 There exists an ideal 𝔞 prime to 6𝔣p such that N𝔞 − ψ(𝔞)^{k} is prime to p.

Proof: It is clear that N𝔞 is congruent to ψ^{k}(𝔞) modulo p for all 𝔞 if and only if χ_{k} is the cyclotomic character. By the Čebotarev density theorem it is sufficient to check this for primes coprime to 6𝔣p since there are only a finite number of exceptions. Hence the lemma follows from Lemma 3.4.2 (and Hypothesis 3.4.4). □

Lemma 3.4.6 C_{∞}^{χ} is free of rank 1. It is generated by (η_{𝔞,n}) where 𝔞 is any ideal of O_{K} prime to 6p𝔣 such that

ψ^{k}(𝔞) ≢ N_{K/ℚ}(𝔞) mod p.

Proof: One may show that for all ideals 𝔞 and 𝔟 of O_{K} not dividing 6p𝔣 and for all n the following relation holds

(η_{𝔞,n})^{F rob_{𝔟}−N𝔟} = (η_{𝔟,n})^{F rob_{𝔞}−N𝔞}  (18)

(see [Rubin2] Theorem 7.4 and Lemma 7.10). On the other hand, by Lemma 3.4.5 we may choose 𝔞 to satisfy ψ^{k}(𝔞) ≢ N𝔞 mod p. This implies that the image of F rob_{𝔞} − N𝔞 is invertible in Λ_{χ}, and hence (by (18) that C_{n}^{χ} is generated as Λ_{χ}- module by η_{𝔞,n} and the image of the p-power roots of unity in K_{n}. Since χ is not the cyclotomic character by Lemma 3.4.2 we deduce that C_{n}^{χ} is generated by η_{𝔞,n}. The norm compatibility of the η_{𝔞,n} shows that the η_{𝔞,n} fit together to give an element η_{𝔞} of C_{∞}. A simple compactness argument then shows that the image of η_{𝔞} generates C_{∞}^{χ}.

Finally we have C_{∞}^{χ} ⊂ U_{∞}^{χ}, the latter being a free Λ_{χ}-module. Hence C_{∞}^{χ} is torsion free and the proof is complete. □

4 Bounding Selmer Groups

4.1 Kato’s Reciprocity Map

Write P = (P_{n})_{n} for the generator of the O_{p}-module T_{E} defined in (17), and from now on fix χ = χ_{k}. In [Kato] Kato considers a homomorphism

l : U_{∞} ⟶ Cotan(Ê)^{⊗k} ⊗ K_{p},

(the tensor products being taken over O_{p}), as the composite of

h : U_{∞}⟶^{α} lim_{⟵} H^{1}(K_{p,n}, ℤ_{p}(1)/p^{n})

⟶^{β} lim_{⟵} H^{1}(K_{p,n}, T_{k}^{∗}(1)/p^{n})⟶^{γ} lim_{⟵} H^{1}(K_{p}, T_{k}^{∗}(1)/p^{n}) ˜ ⟶H^{1}(K_{p}, T_{k}^{∗}(1))

and

exp^{∗} : H^{1}(K_{p}, T_{k}^{∗}(1)) ⟶ H^{1}(K_{p}, V_{k}^{∗}(1)) ⟶ D_{dR}^{0}(V_{k}^{∗}(1)) ˜ ⟶Cotan(Ê)^{⊗k} ⊗ K_{p}.

where

• the inverse limits are with respect to corestriction and the natural projections.

• α is deduced from the Kummer maps for the fields K_{p,n}, β is the inverse limit of cup product with P_{n}^{⊗(−k)}, and γ is corestriction.

One knows that Cotan(Ê) is an invertible O_{p}-module, hence Colie(G)^{⊗k} ⊗ K_{p} is a free K_{p}-module of rank 1.

We shall show that the map

h_{χ} : U_{∞}^{χ} ⟶ H^{1}(K_{p}, T_{k}^{∗}(1))

induced from h is surjective (we shall also exhibit its kernel) and thus obtain the equality

h_{χ}(ker(l_{χ})) = ker(exp^{∗}) = H_{g}^{1}(K_{p}, T_{k}^{∗}(1)),  (19)

where l_{χ} is the restriction of l to U_{∞}^{χ}.

First note that restriction induces an isomorphism

H^{1}(K_{p}, A_{k}) ˜ ⟶H^{1}(K_{∞,p}, A_{k})^{G_{∞}} ˜ ⟶Hom_{ℤ_{p}}(G_{K_{∞,p}}^{ab}, A_{k})^{G_{∞}}.

This follows from the proof of Lemma 6.2 of [Rubin2] and the following lemma .

Lemma 4.1.1 Let Δ ⊂ (O/p^{n})^{×} be the image of

G_{K} ⟶^{ψ^{k}}O_{p}^{×} ⟶ (O_{p}/p^{n})^{×}.

Let Δ act on O_{p}/p^{n} by

z : x ↦ z^{k}x.

Then (under Hypothesis 3.4.4) H^{1}(Δ, O_{p}/p^{n}) = 0.

Proof: Fist note that Δ is not a p-group by the proof of Lemma 3.4.1. Let Δ′ be the prime to p-part of Δ. Then Δ′ is isomorphic to the image of χ. Since O_{p}/p^{n} is a p-group, H^{i}(Δ′, O_{p}/p^{n}) = 0 for i ≥ 1. Furthermore, H^{0}(Δ′, O_{p}/p^{n}) = 0. Indeed, suppose otherwise. Then Δ′ must be killed by k. The order of Δ′ is a multiple of p + 1, so (p + 1) | k. This contradicts Hypothesis 3.4.4.

The inflation/restriction sequence

0 ⟶ H^{1}(Δ/Δ′, (O_{p}/p^{n})^{Δ′}) ⟶ H^{1}(Δ, O_{p}/p^{n}) ⟶ H^{1}(Δ′, O_{p}/p^{n})

now gives us the desired result. □

By local class field theory

G_{K_{∞,p}}^{ab} ≅ U_{∞},

and this isomorphism is G_{∞}-equivariant. Hence

H^{1}(K_{p}, A_{k}) ≅ Hom_{ℤ_{p}}(U_{∞}, A_{k})^{G_{∞}} ≅ Hom_{O_{p}}(U_{∞}^{χ}/A_{k}U_{∞}^{χ}, A_{k}),

where, we recall,

A_{k} = ker(ψ^{k} : Λ_{χ} ⟶ O_{p}).

We deduce an isomorphism of O_{p}-modules

U_{∞}^{χ}/A_{k} ≅ Hom_{O_{p}}(H^{1}(K_{p}, A_{k}), K_{p}/O_{p}),

and hence after applying local Tate duality an isomorphism

t : U_{∞}^{χ}/A_{k}U_{∞}^{χ} ˜ ⟶H^{1}(K_{p}, T_{k}^{∗}(1)).

Proposition 4.1.2 The following diagram commutes:

U_{∞}^{χ} ⟶^{h_{χ}} H^{1}(K_{p}, T_{k}^{∗}(1))
↓  _{id}↓
U_{∞}^{χ}/A_{k} ⟶_{∼}^{t} H^{1}(K_{p}, T_{k}^{∗}(1)).

Proof: Given the definition of Tate duality it suffices to prove

c ∪ h(u) = c^{res}(u)P^{⊗−k}  for all c ∈ H^{1}(K_{p}, A_{k}) and u ∈ U_{∞}^{χ}  (20)

where c^{res} is the image of c in Hom_{ℤ_{p}}(U_{∞}, A_{k})^{G_{∞}}. Note that P occurs in the definition of h, so it is not surprising that it appears in the right hand side of (20).

Let (g(u)_{n})_{n} = βα(u)_{n}, where α and β are the maps occuring in the definition of h. Writing cores_{n} (resp. res_{n}) for corestriction (resp. restriction) on cohomology from K_{n,p} to K_{p}, the projection formula for Galois cohomology tells us that for any n:

c ∪ h(u) = c ∪ cores_{n}(g(u)_{n}) = cores_{n}(res_{n}(c) ∪ g(u)_{n}).  (21)

Let c′_{n} = res_{n}(c) ∪ P_{n}^{⊗−k}, so res_{n}(c) ∪ g(u)_{n} = c′_{n} ∪ α(u)_{n}.

By the continuity of c we may find n_{0} ≥ 1 such that c ∈ im(H^{1}(K_{p}, T_{k}/p^{n_{0}})). From now on let n ≥ n_{0}. Thus c ∈ H^{1}(K_{p}, T/p^{n}).

Lemma 4.1.3 below will tell us that c′_{n} ∪ α(u)_{n} = c′_{n}(u_{n}). Substituting this into (21) we find

c ∪ h(u) = c′_{n}(u_{n}) = c^{res}(u)P^{⊗−k},

and the proof of the proposition is complete.

□

Lemma 4.1.3 Let F be a finite extension of ℚ_{p}, and suppose [F (μ_{p^{n}}) : F] is prime to p. Then the pairing

Φ : H^{1}(F, ℤ/p^{n}) × F^{∗} ⟶ ℤ/p^{n}ℤ
(c, z) ⟼ inv(c ∪ δ(z))

is just Φ(c, z) = c(ρ(z)), where ρ : F^{∗} ⟶ G_{F}^{ab} is the local reciprocity map, δ is the Kummer homomorphism, and we identify H^{1}(F, ℤ/p^{n}) with Hom_{ℤ_{p}}(G_{F}^{ab}, ℤ/p^{n}).

Proof: F (μ_{p^{n}})/F is Galois, and H = G(F (μ_{p^{n}})/F) has order prime to p by hypothesis, so

H^{1}(F, ℤ/p^{n}) = H^{1}(F (μ_{p^{n}}), ℤ/p^{n})^{H}.

Now, a property of the invariant map and restriction implies

Φ(c, z) = (#H)^{−1}inv(res(c) ∪ res(δ(z))),

(this makes sense since #H is prime to p), and the result follows from Proposition 5 in Chapter XIV of [Serre].

□

Proposition 4.1.2 immediately implies (19). Applying Tate local duality, and using Proposition 3.3.4, we deduce

Proposition 4.1.4

H_{f}^{1}(K_{p}, A_{k}) = ker(H^{1}(K_{p}, A_{k}) ⟶ Hom_{ℤ_{p}}(ker(l), A_{k})).
□

4.2 The Selmer group and Kato’s explicit reciprocity law

First note that the choice of Weirstrass model we have made gives us an identification of O(Ê) with O_{p}[[w]], where w = x/y is a local parameter at the identity of E. Since E has supersingular reduction at p we may identify T_{E} with the Tate module of Ê as representation of G_{K_{p}}.

Let Λ_{p,𝔞} ∈ K_{p}((w)) denote the expansion of Λ_{𝔞} in terms of w. One may show explicitly that Λ_{p,𝔞} ∈ O_{p}[[w]]^{×}. With η_{n,𝔞} defined as in Section 3.4, define η_{𝔞} = (η_{n,𝔞})_{n} It follows from the definition of Λ_{𝔞} that Λ_{p,𝔞} is the Coleman power series of η_{𝔞} relative to (P_{n}).

Next we recall the defintion of the Coates-Wiles homomorphisms. Let D = \frac{1}{λ′(w)} \frac{d}{dw} be the logarithmic derivative operator on O_{K_{p}}[[w]], where

λ(w) ∈ w + w^{2}K_{p}[[w]]

is logarithm map of Ê. In fact

λ′(w) ∈ 1 + wO_{p}[[w]],

so it is clear that D does indeed map O_{p}[[w]] into itself. The k^{th} Coates-Wiles homomorphism is defined for k ≥ 1 by

δ : U_{∞} ⟶ O_{p}  (22)
u ↦ (D^{k} log(g_{u}(w))) |_{w=0} .

(We suppress k from the notation since it is fixed). Here log g_{u}(w) is meant in a formal sense: since k ≥ 1,

D^{k} log g_{u}(w) = D^{k−1} (\frac{1}{λ′(w)} \frac{d}{dw} g_{u}) (w).

The Coates-Wiles homomorphisms are compatible with the action of Λ in the following way:

δ(fu) = ψ^{k}(f)δ(u)  (23)

for all f ∈ Λ. Using Lemma 3.4.3 and (23) we find:

Lemma 4.2.1 If u is any element in U_{∞} then δ(Λ_{χ}u) is an ideal in O_{p}. □

The following explicit reciprocity law of Kato gives a cohomological interpretation to δ. See [Kato] for the proof.

Theorem 4.2.2 If

l : U_{∞} ⟶ Cotan(Ê)^{⊗k} ⊗ K_{p}

is the homomorphism defined in section 4.1, then

l(u) = \frac{1}{(k − 1)!}ω^{⊗k} ⊗ ((\frac{d}{ω} )^{k} log(g_{u})) (O_{E}),

where ω is any nonzero element of Cotan(Ê), O_{E} is the identity of the group law on E, and g_{u} denotes the Coleman power series of u relative to P_{n} □

Now, if ω_{E} is the canonical translation invariant holomorphic differential attached to E/K by our choice of Weierstrass equation, then writing ω_{E} in terms of dx and dy, and expanding x and y in power series in w, one finds that

ω_{E} = λ′(w)dw

(see page 8 of [Rubin2]), and this in fact characterises λ as an element of z + z^{2}K_{p}[[w]]. Since λ′(w) ∈ 1 + zO_{p}[[w]], and by its derivation from ω_{E}, we have that λ′(w)dw ∈ Cotan(Ê).

We may thus identify Cotan(Ê)^{⊗k} with O_{p} by choosing (λ′(w)dw)^{⊗k} as a basis. Under this identification the operator \frac{d}{ω} in Theorem 4.2.2 becomes D. Evaluating elements of O(Ê) at the identity correponds to evaluating power series in w at w = 0 under O(Ê) ≅ O_{p}[[w]] since w is a local parameter at the identity. We may thus rephrase the conclusion of Theorem 4.2.2 as

l(u) = \frac{1}{(k − 1)!}δ(u).  (24)

Remark 4.2.3 Notice that we deduce from this that the image of l is a subset of O_{p}, a fact which is by no means obvious from the cohomological definition of l.

We next need to identify the image of l. This is achieved by the following proposition which was shown to us by Coates.

Proposition 4.2.4 If k < p^{2} − 1 then the homomorphism

l : U_{∞} ⟶ K_{p}

has image equal to O_{p}.

Proof: We show that δ has image (k − 1)!O_{p} then appeal to (24).

The tower K_{p,n} is totally ramified and abelian, so by Lubin-Tate theory K_{p,n} corresponds to adjoining to K_{p} the p^{n} torsion of a Lubin-Tate formal group F whose Frobenius lifting is given by the polynomial πX + X^{p^{2}}, for some uniformizer π of K_{p}. By the uniqueness property of the Coleman power series, the construction of the Coates-Wiles homomorphism and Theorem 4.2.2 we see that δ coincides with the Coates-Wiles homomorphism derived from F (see [deShalit]) with respect to the generator v = (v_{n}) corresponding to P under the given isomorphism Ê ≅ F.

Let β be the unique (p^{2} − 1)^{th} root of 1 − π satisfying β ≡ 1 mod p, and let v = (v_{n}) denote a generator of the Tate module of F. The v_{n} is a uniformiser of K_{p,n} by Lubin-Tate theory. A simple calculation shows that

N_{n,n−1}(β − v_{n}) = β − v_{n+1}.

Thus, using the fact that v_{n} is a uniformizer of K_{p,n}, we see u = (β − v_{n}) ∈ U_{∞}. Visibly g_{u}(w) = β − w is the Coleman power series for u relative to F.

If w = ϕ(z) gives the exponential map for F, so ϕ = λ^{−1}, then

\frac{d}{dz} log g_{u}(w) = −\frac{1}{λ′(w)} \frac{−1}{β − w} = −\frac{1}{β} ∑_{r=0}^{∞} \frac{w^{r}}{β^{r}} ,

since 1/λ′(w) = ϕ′(z). On the other hand, we have [ζ]_{F}(w) = ζw for all ζ ∈ μ_{p^{2}−1}, so ϕ(ζz) = ζϕ(z), for all z. Comparing coefficients one both sides of this equality yields

ϕ(z) = z + ∑_{n=1}^{∞} a_{n}z^{1+n(p^{2}−1)}

for some a_{n}. Thus \frac{d}{dz} log g_{u}(w) has coefficients of w^{r} and z^{r} equal for 1 ≤ r < p^{2} −1, and δ(u) = (k − 1)! · unit. The result now follows immediately, since by Lemma 4.2.1 the image of δ must be some ideal in O_{p}. □

We next wish to relate H_{f}^{1}(K, A_{k}) to an Iwasawa module. Firstly, the same proof as in Lemma 6.4 of [Rubin2] tells us that

H_{f}^{1}(K, A_{k})′ ≅ Hom_{ℤ_{p}}(X_{∞}, A_{k})^{G_{∞}},

in view of Lemma 4.1.1.

We now show the existence of a suitable quotient of X_{∞}, with Λ-rank 1.

Lemma 4.2.5 There exists a decomposition U_{∞}^{χ} = V_{1} ⊕V_{2} where V_{1} and V_{2} are free Λ^{χ}-modules of rank 1, l(V_{2}) = 0, and E_{∞}^{χ} ⊄ V_{2}. Furthermore ker(l_{χ}) = A_{k}V_{1} + V_{2}.

Proof: In section 3.4 we remarked that U_{∞}^{χ} is free of rank 2 over Λ_{χ}. Fix a splitting

U_{∞}^{χ} = Λ_{χ}v_{1} ⊕ Λ_{χ}v_{2}.

It follows from Lemma 4.2.1 and (24) that l(Λ_{χ}v_{1}) and l(Λ_{χ}v_{2}) are ideals in O_{p}. Proposition 4.2.4 tells us that l is surjective, so we must have either l(Λ_{χ}v_{1}) or l(Λ_{χ}v_{2}) equal to the whole of O_{p}. Suppose the former, without loss of generality.

We may choose f ∈ Λ_{χ} such that l(v_{2}) = l(fv_{1}). Then if

f_{0} ∈ ker(ψ^{k} : Λ_{χ} ⟶ O_{p})

it follows from (23) and (24) that we have l(V_{2}) = l((f + f_{0})v_{1}). We may choose f_{0} so that, writing g = f + f_{0}, we have E_{∞} ⊄ Λ_{χ}(v_{2} − gv_{1}). The proof is now finished, if we take V_{1} = Λ_{χ}v_{1} and V_{2} = Λ_{χ}(v_{2} − gv_{1}). □

Fix a decomposition of U_{∞}^{χ} as in lemma 4.2.5, and let Ũ = U_{∞}^{χ}/V_{2}. Class field theory gives us a homomorphism of Λ-modules

U_{∞}^{χ} ⟶ X_{∞}^{χ},

so we may define X̃ = X_{∞}^{χ}/im(V_{2}). Since E_{∞}^{χ} ⊄ V_{2} we deduce that Im(V_{2}) has rank 1, and that X̃ is a torsion Λ_{χ}-module. Also, by definition

X_{∞}^{χ}/(A_{k} + im(ker(l_{χ}))) = X̃/A_{k}.

By (10):

H_{f}^{1}(K, A_{k}) = ker (Hom_{ℤ_{p}}(X_{∞}, A_{k})^{G_{∞}} ⟶ \frac{H^{1}(K_{p}, A_{k})}{H_{f}^{1}(K_{p}, A_{k})}) .

Proposition 4.1.4 yields

H_{f}^{1}(K, A_{k}) = ker(Hom_{ℤ_{p}}(X_{∞}, A_{k})^{G_{∞}} ⟶ Hom_{ℤ_{p}}(ker(l), A_{k}))
= Hom_{ℤ_{p}}(X_{∞}^{χ}/(A_{∥}_{k} + im(ker(l))), A)
= Hom_{ℤ_{p}}(X̃, A_{k})^{Γ}.  (25)

We next use the two-variable main conjecture as proved by Rubin to relate the Selmer group to the quotient U_{∞}^{χ}/E_{∞}^{χ}. See [Rubin] and [Rubin2].

Theorem 4.2.6 (The Main Conjecture)

char(A_{∞}) = char(E_{∞}/C_{∞}).
□

A standard exact sequence in global class field theory runs

0 ⟶ E_{∞}/C_{∞} ⟶ U_{∞}/C_{∞} ⟶ X_{∞} ⟶ A_{∞} ⟶ 0,

(see [Rubin2] page 40). We immediately deduce from this an exact sequence

0 ⟶ E_{∞}^{χ}/C_{∞}^{χ} ⟶ Ũ/im(C_{∞}^{χ}) ⟶ X̃ ⟶ A_{∞}^{χ} ⟶ 0.  (26)

Indeed the only question is exactness at E_{∞}^{χ}/C_{∞}^{χ}. This follows from the choice of V_{1} and V_{2}: since E_{∞}^{χ} is torsion-free of rank 1 (it is a submodule of U_{∞}^{χ} which is free), and E_{∞}^{χ} ⊄ V_{2} we must have either that E_{∞}^{χ}/E_{∞}^{χ} ∩ V_{2} is torsion or E_{∞}^{χ} ∩ V_{2} = 0. But U_{∞}^{χ}/V_{2} is torsion-free, so the latter possibility must hold, and exactness follows. From Theorem 4.2.6 and (26) we deduce

char_{Λ_{χ}}(X̃) = char_{Λ_{χ}}(Ũ/im(C_{∞}^{χ})).  (27)

We shall use this equality to deduce that

#H_{f}^{1}(K, A_{k}) = #Hom(Ũ/im(C_{∞}^{χ}), A_{k})^{Γ}.  (28)

More generally, let K_{0} ⊂ L ⊂ K_{∞} be a subextension such that Γ_{L} = G(L/K_{0}) ≅ ℤ_{p}^{d} for d = 1 or 2. Write Λ_{L} = ℤ_{p}[[G(L/K)]].

Given two such subextensions of K_{∞} with F ⊂ L and G(L/F) ≅ ℤ_{p}, the natural projection G(L/K) ⟶ G(F/K) induces a surjective homomorphism of ℤ_{p}-algebras

π_{L/F} : Λ_{L} ⟶ Λ_{F} .

Consider the following condition for torsion Λ_{L}-modules X:

(pseud(L, X)) X has no pseudo-null Λ_{L}-submodules.

The following is due to Perrin-Riou (see [PR] and [Rubin] Lemma 6.2).

Proposition 4.2.7 Suppose that M is a finitely-generated torsion Λ-module such that pseud(L, M) holds whenever L is a ℤ_{p}^{d}-extension of K_{0} in K_{∞}, (d = 1 or 2). Then

#Hom(M, A_{k})^{G_{∞}} = #(O_{𝔭}/(\overline{ψ}^{k}(char_{Λ_{χ}}(M^{χ})))),

where both sides are possibly infinite. □

Corollary 4.2.8

#Hom(X̃, A_{k})^{Γ} = #Hom(Ũ_{∞}/im(C_{∞}), A_{k})^{Γ}.

Proof: By (27) and Proposition 4.2.7 it is enough to show that pseud(L, M) holds for M = X̃ or Ũ/im(C_{∞}^{χ}), and L a ℤ_{p}^{d} extension of K_{0} in K_{∞}, d = 1 or 2. Fix such an L. Now pseud(L, X_{∞}) holds for d = 2 by fundamental results of Greenberg (see [Green]), and V_{2} is free, so pseud(K_{∞}, X̃) holds. Furthermore pseud(L, X̃) holds for d = 1 since (X_{∞})_{L} ≅ X_{∞}(L) as Λ_{L} modules, where X_{∞}(L) denotes the maximal p-unramified abelian p-extension of L. Finally, pseud(L, Ũ/im(C_{∞}^{χ})) holds since Ũ_{∞} and C_{∞}^{χ} are both free Λ^{χ}-modules. □

We may now deduce (28), as promised, from Corollary 4.2.8 and (25). On the other hand l induces a homomorphism

l : Ũ ⟶ O_{p}.

Since l(U_{∞}) = O_{p} and l(V_{2}) = 0 it follows that l(Ũ) = O_{p}. Now l induces

l : Ũ/A_{k} ↪ O_{p},

so this injection is an isomorphism. Hence (28) implies that

#H_{f}^{1}(K, A_{k}) = [O_{p} : l(C_{∞}^{χ})]  (29)

4.3 Connection with L-values

It remains to compare the image of the elliptic units under l with L(\overline{ψ}^{k}, k). This is achieved by a calculation due to Coates and Wiles. The following is Theorem 7.22 of [Rubin2], in light of the reciprocity law and the fact that Λ_{a,𝔞} is the Coleman power series of η_{𝔞}.

Theorem 4.3.1 We have

l(η_{𝔞}) = 12(−1)^{k−1}f^{k}(N𝔞 − ψ(𝔞)^{k})Ω^{−k}L(\overline{ψ}^{k}, k),

where f is such that P_{𝔣} = ξ(Ω/f). □

Corollary 4.3.2

l(C_{∞}^{χ}) = (Ω^{−k}L(\overline{ψ}^{k}, k))O_{p}.

Proof: First note that we may ignore the p-power roots of unity which occur in the definition of C_{n}. Indeed, l_{n} is a homomorphism, factors through C_{n}^{χ}, and since χ is not the cyclotomic character by Proposition 3.4.2 it is trivial on roots of unity. A compactness argument then shows that C_{∞} is generated by the images of η_{𝔞} where 𝔞 runs over ideals prime to 6𝔣p. The image of C_{∞} under l is thus the ideal generated by the image of the η_{𝔞}. We are thus finished by Theorem 4.3.1 (noting that p is prime to 6𝔣) and Lemma 3.4.5.

□

We thus conclude our main result on special values of L-functions.

Theorem 4.3.3 For p satisfying Hypothesis 3.4.4

#H_{f}^{1}(K, A_{k}) = (\frac{L(\overline{ψ}^{k}, k)L(ψ^{k}, k)}{\overline{Ω}^{k}Ω^{k}})_{p} ,

with the understanding that both sides may be simultaneously infinite.

Proof: Substituting the result of Corollary 4.3.2 into (29) we have

#H_{f}^{1}(K, A_{k}) = [O_{p} : (\frac{L(\overline{ψ}^{k})}{Ω^{k}}) O_{p}] .  (30)

Since L(\overline{ψ}^{k}, k) = \overline{L(ψ^{k}, k)} by (8) we have

#H_{f}^{1}(K, A_{k}) = [O_{p} : \frac{L(\overline{ψ}^{k}, k)}{Ω^{k}}O_{p}]

= (\frac{L(\overline{ψ}^{k}, k)L(ψ^{k}, k)}{\overline{Ω}^{k}Ω^{k}})_{p} .

□

Corollary 4.3.4 If k ≥ 2 (and Hypothesis 3.4.4 holds) then H_{f}^{1}(K, A_{k}) is always

finite, and its order is equal to (\frac{L(\overline{ψ}^{k},k)L(ψ^{k},k)}{\overline{Ω}^{k}Ω^{k}})_{p}.

Proof: The Euler product for L(\overline{ψ}^{k}, s) converges absolutely for s > k/2 + 1 yielding the assertion for k > 2. Furthermore it is a classical result following from Tate’s thesis (see [Lang] page 313) that L(ϕ, n) is finite and nonzero whenever ϕ is a nontrivial Gr”ossencharacter of weight 2n and n is an integer. This yields the assertion when k = 2. □

Remark 4.3.5 For k = 1 the Selmer group H_{f}^{1}(K, A_{1}) will be infinite if the Mordell-Weil group of E/K has positive rank, and the converse is true if the Tate-Shafarevich group of E/K has finite p-part.

4.4 Duality

At the beginning of this section we assume only that p is a prime where E has good supersingular reduction. As in earlier sections we deal with the case of V_{k} and the weight −2 representations denoted W_{k} simultaneously. Let U denote any of these representations.

Proposition 4.4.1 For all k ≥ 1 we have

dim H_{f}^{1}(K, U) = dim H_{f}^{1}(K, U^{∗}(1))

Proof: We use the Poitou-Tate global duality sequence

0 ⟶ H_{f}^{1}(K, U) ⟶ H^{1}(G_{S}, U) ⟶ ⨁_{v∈S} \frac{H^{1}(K_{v}, U)}{H_{f}^{1}(K_{v}, U)}

⟶ H_{f}^{1}(K, U^{∗}(1)) ⟶ H^{2}(G_{S}, U) ⟶ ⨁_{v∈S} H^{2}(K_{v}, U) ⟶ 0. (31)

Here S is the finite set of primes consisiting of primes where U is ramified and p. Thus if v ∈ S and v ≠ p, then U^{I_{v}} ≠ U. But U is free of rank 1 over K_{p} and U^{I_{v}} is a K_{p}-subspace of U. This implies that U^{I_{v}} = 0, and

H_{f}^{1}(K_{v}, U) = H^{1}(G_{K_{v}}/I_{v}, U^{I_{v}}) = 0.

On the other hand, by Tate’s calculation of the local Euler-Poincaré characteristic in the unequal residue characteristic case we have

dim H^{1}(K_{v}, U) = dim H^{2}(K_{v}, U),  (32)

while from Proposition 3.1.2 we have

dim \frac{H^{1}(K_{p}, U)}{H_{f}^{1}(K_{p}, U)} = dim H_{f}^{1}(K_{p}, U^{∗}(1)) = 2 + dim H^{2}(K_{p}, U).  (33)

Recall that our convention is to take all dimensions over ℚ_{p}, unless we say otherwise. Equations (32) and (33) together imply

dim (⨁_{v∈S} \frac{H^{1}(K_{v}, U)}{H_{f}^{1}(K_{v}, U)}) = 2 + (⨁_{v∈S} H^{2}(K_{v}, U)) .  (34)

Next, Tate’s calculation of the global Euler-Poincaré characteristic gives

dim H^{1}(G_{S}, U) − dim H^{2}(G_{S}, U) = dim H^{0}(G_{S}, U) + dim U^{∗}(1) = 0 + 2 = 2,

and substituting this as well as equation (34) into the sequence (31) we deduce the result. □

Remark 4.4.2 H_{f}^{1}(K, U) is zero if and only if H_{f}^{1}(K, A_{U}) has a trivial maximal divisible subgroup, and similarly for H_{f}^{1}(K, A_{U}^{∗}(1)).

Proposition 4.4.3 H_{f}^{1}(K, A_{U}) is finite if and only if H_{f}^{1}(K, A_{U}^{∗}(1)) is finite. in this case

#H_{f}^{1}(K, A_{U}^{∗}(1)) = #H_{f}^{1}(K, A_{U}) < ∞.

Furthermore if k ≥ 2 then both H_{f}^{1}(K, A_{k}) and H_{f}^{1}(K, A_{k}^{∗}(1)) are finite (see Section 5 for the situation for W_{k} and W_{k}^{∗}(1)).

Proof: H_{f}^{1}(K, A_{U}) and H_{f}^{1}(K, A_{U}^{∗}(1)) are co-finitely generated. This follows from the fact that H^{1}(G_{S}, U) has finite dimension.

Suppose H_{f}^{1}(K, A_{U}) is finite. Then by Remark 4.4.2 H_{f}^{1}(K, U) = 0. By Proposition 4.4.1 H_{f}^{1}(K, U^{∗}(1)) = 0, so H_{f}^{1}(K, A_{U}^{∗}(1)) is also finite. This argument is reversible, and gives the first part of the Proposition.

On the other hand, if k ≥ 2 then H_{f}^{1}(K, A_{k}) is finite by Corollary 4.3.4 and Corollary 4.3.4, so by the first part H_{f}^{1}(K, A_{k}^{∗}(1)) is also finite.

If both H_{f}^{1}(K, A_{U}) and H_{f}^{1}(K, A_{U}^{∗}(1)) are finite then we may utilise the generalised Cassels-Tate pairing of [Flach] which exhibits the two groups as duals of one another. The result is immediate. □

Proposition 4.4.4 For k ≥ 1 we have

(\frac{L(\overline{ψ}^{k}, k)}{Ω^{k}})_{p} = (\frac{(2π)^{k−1}}{(k − 1)!} \frac{L(ψ^{k}, 1)}{Ω^{k}})_{p} .

Proof: This follows by substituting s = k into the functional equation (7), and noting that since p is inert in K/ℚ, ϵ(\overline{ψ^{k}}, k) is prime to p. □

Corollary 4.4.5 For k ≥ 1 we have

#H_{f}^{1}(K, A_{k}^{∗}(1) = (\frac{(2π)^{2k−1}}{(k − 1)!^{2}} \frac{L(ψ^{k}, 1)}{Ω^{k}} \frac{L(\overline{ψ}^{k}, 1)}{\overline{Ω}^{k}})_{p} .

If k ≥ 2 then both sides are finite.

Proof: This follows from Propositions 4.4.3 and 4.4.4. □

4.5 Descent to ℚ

In this section we indicate how our results may be descended to ℚ. In other words we show how to prove analogous formulae over ℚ, under the assumption (which we make from this point on) that our elliptic curve E is defined over ℚ.

A special case of a very general result is

Lemma 4.5.1 Let c denote a generator for G(K/ℚ). We have

H_{f}^{1}(K, res(A_{U})) = H_{f}^{1}(K, res(A_{U}))^{c=1} ⊕ H_{f}^{1}(K, res(A_{U}))^{c=−1}.

Here res(A_{U}) denotes the restriction of the G_{ℚ}-representation A_{U} to G_{K}.

Proof: This follows because the Selmer groups are p-torsion, and p is odd. □

Lemma 4.5.2 Suppose H_{f}^{1}(K, res(A_{U})) has finite order. Then

#H_{f}^{1}(K, res(A_{U}))^{c=1} = #H_{f}^{1}(K, res(A_{U}))^{c=−1}.

Proof: First note that p ∤ d_{k}. This is because p divides d_{K} if and only if E has bad reduction at p, and we have assumed that E has good reduction at p. The complex multiplication by \sqrt{d_{K}} acts on A_{U}, hence by functoriality, and since p ∤ d_{K} it induces an involution of H_{f}^{1}(K, res(A_{U})) which swaps the two eigenspaces for c (since G_{ℚ} acts K-semi-linearly). □

Lemma 4.5.3

H_{f}^{1}(K, res(A_{U}))^{c=1} = H_{f}^{1}(ℚ, A_{U}).

Proof: This follows easily from the fact that p is odd, and G(K/ℚ) has order 2. □

Remark 4.5.4 Notice that we also have

H_{f}^{1}(ℚ, tw(A_{U})) = H_{f}^{1}(K, tw(A_{U}))^{c=1}
= (tw(H_{f}^{1}(K, A_{U})))^{c=1}
= H_{f}^{1}(K, A_{U})^{c=−1},  (35)

where tw denotes a twist by the Galois character associated to K. Thus by Lemmas 4.5.2, 4.5.3, and by (35) we obtain

H_{f}^{1}(ℚ, A_{U}) ≅ H_{f}^{1}(ℚ, tw(A_{U})).

Let Ω_{+} denote the real period of E. In other words, if ω is the canonical differential attached to our choice of Weierstrass equation for E, then Ω_{+} is the integral of ω over the connected component of the identity on the real points of E. Equivalently, if L is the lattice of periods of E/ℂ, Ω_{+} is the minimal positive real lattice point (it is well known that L is fixed by complex conjugation, so such points exist). There is the following relation between Ω and Ω_{+}.

Lemma 4.5.5 Ω_{+} = zΩ, where z ∈ O_{K} is prime to 6\sqrt{d_{K}}.

Proof: Let Ωz be the real period of L. Then Ωz = \overline{Ω}\overline{z}. Hence Ω = \overline{Ω}w for some w ∈ K. Taking norms we find N_{K/ℚ}(w) = 1, so w is a unit in O_{K}. We have \overline{z} = wz. We consider cases.

If w = 1 then z is real. By minimality z = 1, and Ω = Ω_{+}. If w = −1 then z is purely imaginary. Hence by minimality z = \sqrt{d_{K}}.

If w = ±i, we see that K = ℚ(i) and check (using minimality) that z | 2. Similarly, if w^{3} = 1, then K = ℚ(\sqrt{−3}), and z | 3. □

Theorem 4.5.6 If k ≥ 1 then

#H_{f}^{1}(ℚ, A_{k}) = (\frac{L(\overline{ψ}^{k}, k)}{Ω_{+}^{k}})_{p}  (36)

and

#H_{f}^{1}(ℚ, A_{k}^{∗}(1)) = (\frac{L(ψ^{k}, k)}{Ω_{+}^{k}})_{p} .  (37)

Furthermore, if k ≥ 2 then all expressions are finite.

Proof: By Lemmas 4.5.1, 4.5.2, and 4.5.3 we have that

#H_{f}^{1}(K, res(A_{k})) = (#H_{f}^{1}(ℚ, A_{k}))^{2} .

Hence (36) follows by substituting these and (9) into Theorem 4.3.3. Note that after Lemma 4.5.5 and by our hypothesis on p we may replace Ω by Ω_{+}. The rest follows similarly, using Corollary 4.4.5. □

5 Application to Chow Groups

Though results on the special values of L-functions are interesting in their own right, it turns out they they have applications in the study of cycle class maps. Langer and Raskind [LR] are able to show the finiteness of CH_{0}(E × E){p}, for E/ℚ an elliptic curve with complex multiplication by O_{K} having good reduction at p > 3, using Corollary 4.4.5 for the supersingular reduction case. In this section we show how our results may be used together with fundamental results of Nekovář [N2] to provide some information about higher cycle class maps for the products E^{d}, d ≥ 1.

5.1 Results modulo torsion

Let X be a smooth projective scheme over a number field F. There is a cycle class map into continuous étale cohomology (see [Jan]) for any prime p

cl_{X} : CH^{i}(X) ⊗ ℚ_{p} ⟶ H^{2i}(X_{et}, ℚ_{p}(i)).

Writing V^{i} = V^{i}(X) = H^{i}((X ⊗ \overline{F})_{et}, ℚ_{p}) one has the convergent Hoschild-Serre spectral sequence

E_{2}^{a,b} = H^{a}(F, V^{b}(X)(i)) ⇒ H^{a+b}(X_{et}, ℚ_{p}(i))

inducing a filtration on its abutment. In fact it degenerates at the E_{2} term, so

gr^{j}H^{2i}(X_{et}, ℚ_{p}(i)) = H^{j}(F, V^{2i−j}(i)).

It is expected that there is a motivic version of the Hoschild-Serre spectral sequence yielding a filtration on rational Chow groups (motivic cohomology groups) strictly compatible with the above filtration on l-adic cohomology, and that the global dimension of the category of mixed motives over F is 1. This would imply that

Im cl_{X} ∩ F^{2}H^{2i}(X_{et}, ℚ_{p}(i)) = 0.  (38)

One may attempt to prove (38) independently of the above motivic considerations, and we may make a first step by reducing it to a certain local-global principle.

Suppose X is a smooth projective model of X over Spec O_{F} [\frac{1}{N}] for some (squarefree) integer N, and write

j : X ↪ X

for the inclusion. Write S = {v | pN}. There are versions of the above constructions for X and one may show that

Im(cl_{X}) ∩ F^{2}H^{2i}(X_{et}, ℚ_{p}(i)) = j^{∗}(H),

where

H = (Im(cl_{X}) ∩ F^{2}H^{2i}(X_{et}, ℚ_{p}(i))) .

The following is a special case of a fundamental theorem proved in [N2]. Fix i and put V = V^{2i−2}, to ease notation.

Theorem 5.1.1 Suppose that X has potentially good reduction at every prime ideal of F. Then

Im(cl_{X}) ∩ F^{2}H^{2n}(X_{et}, ℚ_{p}(i))

is a subquotient vector space of

K′ = ker (α : H^{2}(G_{F,S}, V) ⟶ ⨁_{v∈S} H^{2}(F_{v}, V)) .

On the other hand global Poitou-Tate duality gives K′ as the dual of

K = ker (β : H^{1}(G_{F,S}, V^{∗}(1)) ⟶ ⨁_{v∈S} H^{1}(F_{v}, V^{∗}(1))) ,

and it is conjectured that K always vanishes. While this local-global principle seems to be hard to prove in general, we may sometimes prove the triviality of

H_{f}^{1}(F, V^{∗}(1)) = ker (H^{1}(G_{F,S}, V^{∗}(1)) ⟶ ⨁_{v∈S} \frac{H^{1}(F_{v}, V^{∗}(1))}{H_{f}^{1}(F_{v}, V^{∗}(1))}) ,

an a priori larger group. The dimension of H_{f}^{1}(F, V^{∗}(1)) is conjecturally controlled by the order of zero of the L-function L(V, s) at s = 0 (at least if H^{0}(F, V^{∗}(1)) = 0). Of course L(V, s) may vanish at s = 0, but there are interesing situations where one may show that it does not vanish, and in his case one might try to prove the part of the Bloch-Kato conjecture which tells us that H_{f}^{1}(F, V^{∗}(1)) is trivial, hence yielding (38).

From now on let F = K, the quadratic imaginary field considered previously. Put X/K = (E/K)^{d}, the self product of d copies of E/K, where E/K is the elliptic curve with complex multiplication considered in previous chapters, and suppose for the moment that E is defined over ℚ. The Künneth formula gives V as a direct sum of groups of representations of the form V^{1}(E)^{⊗_{ℚp}2i−2−2k}(−k) for varying k. The next step is thus to understand the representations V^{1}(E)^{⊗_{ℚp}j} in terms of the V_{k} considered in earlier sections. We work over K, since the results of (4.5) will yield results over ℚ.

As a representation of G_{K}, some linear algebra yields

V^{1}(E)^{⊗_{ℚp}2} = ((Λ_{ℚ_{p}}^{2}V^{1}(E)) ⊗ K_{p}) ⊕ Sym_{K_{p}}^{2}V^{1}(E),

which by the Weil pairing is

K_{p}(−1) ⊕ V^{1}(E)^{⊗_{Kp}2} = K_{p}(−1) ⊕ V_{2}(−2).

Here we have identified V^{1}(E)(1) with the rational Tate module V_{1}. Hence, again as G_{K} representations, we find that V^{1}(E)^{⊗_{ℚp}2i−2−2k}(i − k) is a direct sum of representations of the form

W_{m} ≅ V_{2}(−2)^{⊗_{Kp}m}(1 + m),

for 0 ≤ m ≤ i − k − 1. Taking the Tate twisted dual, we find (using the Weil pairing here) that we need to bound the Selmer groups for the representations

W_{m}^{∗}(1) ≅ V_{2}(E)^{⊗_{Kp}m}(−m).

In general all these representations will occur with various (nonzero) multiplicities, but it is irrelevant to us what these multiplicities are.

In [Han], Han considers the Tate twisted representations

V_{j,k} = V_{1}(E)^{⊗_{Kp}(j+k)}(−j)  j ≥ 0  k > j + 1,

whose Selmer group should conjecturally be related to the L-value

L(\overline{ψ}^{j+k}, k).

Here E is as above. In the case j = 0 we find ourselves in the situation discussed in the first four sections of this article. Han is able to prove the following.

Theorem 5.1.2 Let T_{j,k} be the canonical lattice in V_{j,k}. Let p be a prime such that

• p > 3,

• ψ^{k}\overline{ψ}^{−j} is nontrivial when restricted to the decomposition group of K(E[p])/K at each prime above p.

Under these hypotheses, the Selmer group H_{f}^{1}(K, V/T) is finite with order equal to

((k − 1)!\frac{(2π)}{\sqrt{d_{K}}} \frac{L(\overline{ψ}^{j+k}, k)}{Ω_{∞}^{j+k}})_{p} .

Remark 5.1.3 Note that here p may be a prime of ordinary reduction, and possibly bad reduction for E. Actually Han goes further to show the Bloch-Kato conjecture holds for the corresponding motive. Also note that we have verified in our situation that the second hypothesis above always holds for p > k, (with j = 0, of course).

Remark 5.1.4 If p is prime to 6𝔣 then ψ is surjective. If p is supersingular then the condition on ψ in the statement of the theorem will be valid for p such that p^{2} − 1 ∤ n + 1 + p(n − 1) for 0 ≤ n ≤ i − 1 (see Lemma 3.4.2). In particular this will hold for p > 2d + 1.

If p is ordinary and prime to 6𝔣, then the condition becomes n + 1 + p(n − 1) ≡ 0 mod p − 1, i.e p − 1 ∤ 2n. Again this will hold for p > 2d + 1.

Recall that we are interested in a representation of weight 0. Han’s are of weight less than or equal to −2. On the other hand V_{m−1,m+1}^{∗}(1) = V_{1}(E)^{2m}(−m). We may thus use Proposition 4.4.1, and Han’s result for V_{m−1,m+1}, together with the observation we have made before that H_{f}^{1}(K, A_{U}^{∗}(1)) is finite if and only if H_{f}^{1}(K, V_{U}^{∗}(1)) is trivial. We deduce the following, which is our main theorem modulo torsion. See the Appendix for remarks on the proof in the case of an ordinary prime.

Theorem 5.1.5 Suppose E is an elliptic curve over ℚ, with good reduction at a prime p > 3. Let i be a strictly positive integer. Assume that

• p^{2} − 1 ∤ n + 1 + p(n − 1) if E has supersingular reduction at p,

• p − 1 ∤ 2n if E has ordinary reduction at p,

for all 0 ≤ n ≤ i − 1. Then

Im(cl_{X}(CH^{i}(X)) ∩ F^{2}H^{2i}(X_{et}, ℚ_{p}(i)) = 0,

where X is the d-fold product of E. The same is true for E ⊗_{ℚ} K.

□

5.2 Integral results

Let our notation be as at the beginning of Section 5.1. Most of Section 5.1 carries through to the integral situation, given that the theorem of Han actually yields the finiteness of an integral Selmer group, not just the vanishing of H_{f}^{1}(K, V) which is what was used in the last section. There are some points which require modification, however, and in fact our final theorem is stronger than the analgue of Theorem 5.1.5.

There is an integral cycle class map into continuous étale cohomology

cl_{X} : CH^{i}(X) ⟶ H^{2i}(X_{et}, ℤ_{p}(i)).

Writing T^{i}(X) = H^{i}((X ⊗ \overline{F})_{et}, ℤ_{p}) one has the Hoschild-Serre convergent spectral sequence

E_{2}^{a,b} = H^{a}(G_{F} , T^{b}(X)(i)) ⇒ H^{a+b}(X_{et}, ℤ_{p}(i)).

This degenerates at the E_{2} term up to finite groups, and in any case induces a filtration on its abuttment. It is expected that

Im(cl_{X}) ∩ F^{2}H^{2i}(X_{et}, ℤ_{p}(i))

is always finite. Suppose that T^{b} is torsion free for all b. Then by the Weil conjectures E_{2}^{0,b} = 0 for all b ≠ 2i. Since cd_{p}(G_{F}) = 2, we deduce

F^{2}(H^{2i}(X_{et}, ℤ_{p}(i)) = gr^{2}(H^{2i}(X_{et}, ℤ_{p}(i))) = H^{2}(G_{F} , T^{2i−2}(i)).

One may as before show that

Im(cl_{X}) ∩ F^{2}H^{2i}(X_{et}, ℤ_{p}(i)) = j^{∗}(H),

where

H = (Im(cl_{X}) ∩ F^{2}H^{2i}(X_{et}, ℤ_{p}(i))) .

On examining the local situation one finds that if p > max{2, 2i + 1, dim X }, and if at every prime v ∈ S X_{v} achieves good reduction after base change to a field of degree prime to p over F_{v} then

Im(cl_{X_{v}}) ∩ F^{2}H^{2i}((X_{v})_{et}, ℤ_{p}(i)) = 0

for every v ∈ S. There are two cases. If v | p then this is Proposition III.5.5 in [N2]. If v ∤ p then this is well known (see Proposition II.2.8 in [N2]).

Remark 5.2.1 In fact it follows from III.5.6 in [N2] that we may also take p = 2i+1 for the choice of X we shall shortly restrict to. However the conditions in Han’s result require us to restrict to p > 2i + 1.

Suppose from now on that we are in the case where the Hoschild - Serre spectral sequence degenerates (not just modulo finite groups). This will be the case later since all our cohomology groups will be torsion free. It follows from the compatibility of all the above constructions with base change that

H ↪ ker (α : H^{2}(G_{F,S}, T) ⟶ ⨁_{v∈S} H^{2}(F_{v}, T)) ,

where T = T^{2i−2}(i). The Poitou-Tate duality sequence gives ker(α) as the dual of

K = ker (β : H^{1}(G_{F,S}, T^{∗}(1) ⊗ ℚ_{p}/ℤ_{p}) ⟶ ⨁_{v∈S} H^{1}(F_{v}, T^{∗}(1) ⊗ ℚ_{p}/ℤ_{p})) .

Ideally one would like to prove that K was always finite, but this seems to be hard. Our approach is to estimate K by a Selmer group which we may be able to bound in terms of an L-value. Of course this does not work if the L-value is zero (but see [Ot]).

As before we put F = K, and X/K = (E/K)^{d}. The linear algebra considered in the last part works integrally, so long as we take p > d (there are denominators involved in the various splittings of the representations).

The duality in Section 4.4 was dealt with on the integral level, so we may use it here, and similarly for the descent from K to ℚ. We have proved all but the last part of the following.

Theorem 5.2.2 Let d be a positive integer, and let p > 3 be a rational prime such that

• p > 2i + 1 (coming from the local result of Nekovář and the local condition of Han),

• p > d,

• E has good reduction at p.

Under these conditions

Im(cl_{X}(CH^{i}(X)) ∩ F^{2}H^{2i}(X_{et}, ℤ_{p}(i))

is finite, and is trivial for almost all primes. In fact

Im(cl_{X}(CH^{i}(X)))_{tors}

is finite for p as above, and trivial if p also satisfies

p ∤ (2π)^{j} \frac{L(ψ^{2j}, −j)}{Ω^{2j}}.

Proof: We need to prove only the final statement. We show the equality

H^{2i}(X_{et}, ℤ_{p}(i))_{tors} = (F^{2}H^{2i}(X_{et}, ℤ_{p}(i)))_{tors}.

This together with the first statement will yield the final statement of the theorem. It is sufficient to show that both gr^{0}H^{2i}(X_{et}, ℤ_{p}(i)) and gr^{1}H^{2i}(X_{et}, ℤ_{p}(i)) are torsion free. But

gr^{0}H^{2i}(X_{et}, ℤ_{p}(i)) ≅ (H^{2i}(\overline{X}_{et}, ℤ_{p}(i)))^{G_{ℚ}},

which is trivial by Künneth and because the Tate module of E is torsion free. Furthermore,

gr^{1}H^{2i}(X_{et}, ℤ_{p}(i)) ≅ H^{1}(ℚ, H^{2i−1}(\overline{X}_{et}, ℤ_{p}(i))),

and considering the long exact sequence corresponding to

0 ⟶ T ⟶ V ⟶ V/T ⟶ 0

we see that

H^{1}(ℚ, H^{2i−1}(\overline{X}_{et}, ℤ_{p}(i)))_{tors} ≅ H^{0}(ℚ, V/T),

where T ≅ H^{2i−1}(\overline{X}, ℤ_{p}(i)), and V = T ⊗_{ℤ_{p}} ℚ_{p}. the group H^{0}(ℚ, V/T) is trivial by the surjectivity of ψ, and our conditions on p.

5.3 Remarks on Ordinary Primes

We restricted to the case of supersingular reduction in the first four sections of this paper only because the case of ordinary reduction was dealt with by Li Guo in [LiGuo]. The proofs of Theorems 5.2.2 and 5.1.5 are, if anything, easier in the ordinary case. We give a few remarks on the differences in this situation for the interested reader.

If E has good ordinary reduction at p then p splits in K/ℚ,

(p) = 𝔭\overline{𝔭}.

Write K_{𝔭} and K\overline{𝔭} for the completion of K at these primes. Both fields are isomorphic to ℚ_{p}. Let π be a uniformiser of K_{𝔭}. The Tate module of E, is still free of rank 1 over K ⊗ ℚ_{p}, and this induces a decomposition of G_{K} representations V ≅ V_{π} ⊕ V\overline{π}. Being subrepresentations of V, both V_{π} and V\overline{π} are crystalline when restricted to G_{ℚ_{p}}.

In the Poitou-Tate sequence there are now two local conditions for v | p. We calculate that for the field K_{𝔭},

dim D_{dR}^{0}(V_{π}) = 0, dim D_{dR}^{0}(V\overline{π}) = 1,

using the fact that V\overline{π} is unramified. By Proposition 3.1.2 this yields

dim H_{f}^{1}(K_{𝔭}, V_{π}) = 1, dim H_{f}^{1}(K\overline{𝔭}, V_{π}) = 0.

The analogue of Proposition 4.4.1 follows immediately.

Everything else works as before: Han’s result holds for ordinary primes. As already remarked, the bounding of the Selmer group in this case is originally due to Li Guo. The statement of Li Guo’s result is the following (see [LiGuo]).

Theorem 5.3.1 Let E be as before, and p be a prime of good ordinary reduction for E. Let V_{j,k} be as before, and suppose that p > k + 1, and 0 < j < k. Then

#(S^{str}(K, V_{j,k}) = ((2π)^{j} \frac{L(ψ^{j+k}, k)}{Ω_{∞}^{j+k}} \frac{L(\overline{ψ}^{j+k}, k)}{\overline{Ω}_{∞}^{j+k}})_{p} .

Here S^{str}(K, A_{j,k}) denotes the strict Selmer group of Greenberg. Han uses the same Selmer groups as us (and Bloch-Kato), but see ([Flach]) for a comparison of these approaches.

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