For brevity, we only discuss here the main conjecture for elliptic curves over Q over the field generated by the coordinates of all p-power division points on the curve, where p is a prime ≥ 5 of good ordinary reduction. However, it will be clear that a similar conjecture can be formulated in great generality for motives overp-adic Lie extensions of number fields which contain the cyclotomic Zp-extension of the base field.
If F is a finite extension of Q , we write Fcyc for the cyclotomic Zp-extension ofF, and putΓF =G(Fcyc/F). LetE be an elliptic curve defined over Q , and Ep∞ the group of all p-power division points on E. We define
F∞ =Q(Ep∞).
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By the Weil pairing, F∞⊃Q(µp∞), whereµp∞ denotes the group of all p-power roots of unity. Hence F∞ contains Qcyc, and we define
G=G(F∞/Q), H=G(F∞/Qcyc), Γ =G(Qcyc/Q).
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Thus Gis a compact p-adic Lie group to which the abstract theory developed in the first four sections of this paper applies. In fact, the structure of G is well known. In-deed, G can be identified with a closed subgroup of GL2(Zp) = Aut(Ep∞). When E admits complex multiplication, G has dimension 2, and when E does not admit com-plex multiplication, G has dimension 4 [30].
If L is any algebraic extension of Q , we recall that the classical Selmer group S(E/L)is defined by
S(E/L)=Ker
H1(L,Ep∞)→
w
H1(Lw,E(L¯w)) ,
wherew runs over all non-archimedean places of L, and Lw denotes the union of the completions at w of all finite extensions of Q contained in L. We write
X(E/L)=Hom(S(E/L),Qp/Zp)
for the compact Pontrjagin dual of the discrete abelian groupS(E/L). We shall mainly be interested in the case in which Lis Galois over F, in which case bothS(E/L) and X(E/L) have a natural left action of G(L/F), which extends to a left action of the whole Iwasawa algebra Λ(G(L/F)). It is easy to see that X(E/L) is always a finitely generated Λ(G(L/F))-module.
We impose for the rest of the paper the following
Hypotheses on p. — p ≥5 and E has good ordinary reduction at p.
The condition p ≥ 5 guarantees that G has no element of order p, since G is a closed subgroup of GL2(Zp). It is a basic question as to when, assuming our hy-potheses on p, the dual Selmer group X(E/F∞) belongs to the category MH(G), or equivalently is S∗-torsion, where S∗ is the Ore set defined in § 3.
Conjecture 5.1. — Assuming our hypotheses on p, X(E/F∞) belongs to the category MH(G).
We now briefly discuss what is known at present about Conjecture 5.1. Indeed, all we know is related to the following much older conjecture of Mazur [23].
Conjecture 5.2. — Assume E has good ordinary reduction at p. For each finite extension F of Q , X(E/Fcyc) is Λ(ΓF)-torsion.
One might hope that Conjecture 5.1 is equivalent to knowing Conjecture 5.2 for all finite extensionsF of Q contained in F∞. At present, we can only prove some partial results in this direction, which we now describe. If Fis a finite extension of Q contained in F∞, we put
GF =G(F∞/F), HF =G(F∞/Fcyc).
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Let Xbe a finitely generatedΛ(G)-module. IfLis a finite extension of Q inF∞ such that GL is pro-p, the µ-invariant of X viewed as a Λ(GL)-module, is defined in [19]
and [36]. Similarly, if Z is any finitely generated Λ(ΓF)-module, we write µΓF(Z) for its classical µ-invariant. Also, for any field K in F∞, we define
Y(E/K)=X(E/K)/X(E/K)(p).
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Lemma 5.3. — Assume that X(E/F∞)belongs to MH(G). Then, for each finite extension Fof Q contained in F∞, we have(i)X(E/Fcyc)isΛ(ΓF)-torsion; (ii)Hi(HF,X(E/F∞))=0 for all i ≥ 1; (iii) Hi(HF,Y(E/F∞)) is finite for i = 1,2, and zero for i > 2. Moreover, we have µGL(X(E/F∞)) =µΓL(X(E/Lcyc)) for each finite extension L of Q in F∞ such that GL
is pro-p.
Proof. — We assume that X(E/F∞) belongs to MH(G), so thatY(E/F∞)is a fi-nitely generated Λ(H)-module. Take F to be any finite extension of Q contained in F∞. Then Y(E/F∞)HF is clearly a finitely generated Zp-module, because Y(E/F∞) is then a finitely generated Λ(HF)-module. Now we have the commutative diagram with exact rows
X(E/F∞)HF −−−→ Y(E/F∞)HF −−−→ 0
rF sF
X(E/Fcyc) −−−→ Y(E/Fcyc) −−−→ 0,
where rF is the dual of the restriction map tF from S(E/Fcyc) to S(E/F∞)HF, and sF is the corresponding induced map. We claim that the cokernel of rF is finite. In-deed, the cokernel of rF is dual to the kernel of the restriction map tF, and Ker(tF) is contained in H1(HF,Ep∞). But this latter group is proven to be finite in [11]. Since Coker(rF) is finite, it follows that Coker(sF) is also finite. As was remarked above, Y(E/F∞)HF is a finitely generatedZp-module, and so we see that Y(E/Fcyc)is a finitely generated Zp-module. Thus X(E/Fcyc) is Λ(ΓF)-torsion, proving (i). To prove the re-mainder of Lemma 5.3, we invoke the arguments of [9], and do not repeat the de-tailed proofs of these here. SinceX(E/Fcyc)is Λ(ΓF)-torsion, Lemma 2.5 of [9] shows that H1(HF,X(E/F∞))= 0. As we have shown X(E/Fcyc) is Λ(ΓF)-torsion for all F, Lemma 2.1 and Remark 2.6 of [9] prove that Hi(HF,X(E/F∞))=0 for all i ≥2, es-tablishing (ii). Turning to (iii), an entirely similar argument to that given in the proof of Proposition 2.13 of [9] gives that H3(HF,Y(E/F∞)) =0, and that Hi(HF,Y(E/F∞)) (i = 1,2) are killed by a power of p. Hence, as the Hi(HF,Y(E/F∞)) (i ≥ 0) are finitely generated Zp-modules, we conclude that Hi(HF,Y(E/F∞)) must be finite for i=1,2. The final assertion of Lemma 5.3 now follows from formula (25) of Proposi-tion 2.13 of [9], on noting that theµ-invariant of theΛ(ΓL)-module H0(HL,Y(E/F∞)) is zero, because this module is finitely generated over Zp. This completes the proof of Lemma 5.3.
Lemma 5.4. — Assume thatX(E/Fcyc)is Λ(ΓF)-torsion for all finite extensions F of Q inF∞. Suppose, in addition, that there exists a finite extension Lof Q contained in F∞ satisfying:
(i) GL is pro-p, (ii) µGL(X(E/F∞)) = µΓL(X(E/Lcyc)), and (iii) H1(HL,Y(E/F∞)) is finite. Then X(E/F∞)belongs to MH(G).
Proof. — We first observe that H0(HL,Y(E/F∞)) is a finitely generated torsion Λ(ΓL)-module, assuming the hypotheses of the lemma. Indeed, H0(HL,Y(E/F∞)) is a quotient ofH0(HL,X(E/F∞)), and we claim that this latter module is a finitely gen-erated torsion Λ(ΓL)-module. This is because we are assuming that the finitely gener-atedΛ(ΓL)-module X(E/Lcyc) is Λ(ΓL)-torsion, and it is shown in [8] that the kernel of the natural map
rL:X(E/F∞)HL →X(E/Lcyc),
which is the dual of the restriction map, is a finitely generated Zp-module. Hence Ker(rL)is a finitely generated torsion Λ(ΓL)-module, and our claim follows. The del-icate part of the proof is to now show that
µΓL(H0(HL,Y(E/F∞)))=0. (100)
Indeed, (100) implies immediately that Y(E/F∞) is a finitely generated Λ(H)-module, completing the proof of the lemma. This is because (100) shows thatH0(HL,Y(E/F∞)) is a finitely generatedZp-module, and hence, asHL is pro-p, Nakayama’s lemma gives that Y(E/F∞) is finitely generated over Λ(HL). To prove (100), we invoke the full force of Proposition 2.13 of [9]. As remarked earlier, our assumption that X(E/Fcyc) is Λ(ΓF)-torsion for all finite extensions F of Q in F∞ implies that all the hypothe-ses of Proposition 2.13 of [9] are valid. Hence formula (25) of Proposition 2.13 of [9]
holds. Inserting conditions (ii) and (iii) in formula (25) of [9], we conclude that theµΓL -invariants of H0(HL,Y(E/F∞)) and H2(HL,Y(E/F∞)) must add up to zero. Thus, as both are non-negative integers, both of these µ-invariants must be zero, proving (100) in particular.
Corollary 5.5. — Assume that p ≥ 5, E has good ordinary reduction at p, and E ad-mits complex multiplication. ThenX(E/F∞)belongs to MH(G) if and only if there exists a finite extension L of Q in F∞ such that GL is pro-p and µΓL(X(E/Lcyc))=0.
Proof. — It is a well known consequence of the proof of the two variable main conjecture for E over F∞ (see [27]) that X(E/Fcyc) is Λ(ΓF)-torsion for every finite extension F of Q in F∞. Take L to be any finite extension of Q contained in F∞ such that GL is pro-p. Then it is well known that the thesis of Schneps [28] implies that µGL(X(E/F∞)) = 0. This in turn implies that X(E/F∞)(p) = 0 because X(E/F∞) has no non-zero Λ(GL)-pseudo-null submodule by [26]. We then have Hi(HL,Y(E/F∞)) = 0 for all i ≥ 1 because Y(E/F∞) = X(E/F∞). The assertion of the corollary is now clear from Lemmas 5.3 and 5.4.
The following is the one practical criterion we know at present for proving in some concrete examples that X(E/F∞) belongs to MH(G).
Proposition 5.6. — Assume that there exists a finite extension L ofQ contained inF∞, and an elliptic curve E defined over L, as follows: (i) GL is pro-p, (ii) E is isogenous to E over L, and (iii) X(E/Lcyc) is finitely generated Zp-module. Then X(E/F∞) belongs to MH(G).
It is very probable that the hypotheses of Proposition 5.6 are valid for most el-liptic curvesEover Q and most primesp of good ordinary reduction. However, in our present state of knowledge, condition (iii) is difficult to verify in numerical examples.
Here is an example of one isogeny class of curves to which it can be applied.
Example. — Consider the three elliptic curves over Q of conductor 11, which we denote by Ei (i = 0,1,2). Here E1 is given by (70), and the other two are given by the equations
E0:y2+y=x3−x2−10x−20
E2:y2+y=x3−x2−7820x−263580.
All three curves are isogenous over Q , and so the field F∞ is the same for the three curves and all primes p. Taking p = 5, and L = Q(µ5), it is well known [16] that GL is pro-5 and X(E1/Lcyc)=0. Hence Proposition 5.6 shows that X(Ei/F∞)belongs to MH(G) for p =5 and i =0,1,2. At present, we do not know how to prove that X(Ei/F∞) belongs to MH(G) for any good ordinary prime p > 5.
We now prove Proposition 5.6. Pick an isogeny f : E → E, which is defined over L. Note that F∞=L(Ep∞). The natural homomorphism
rL:X(E/F∞)HL →X(E/Lcyc),
which is the dual of the restriction map, has a kernel which is a finitely generated Zp-module (see [8]). Applying Nakayama’s lemma, we deduce from (i) and (iii) that X(E/F∞) is finitely generated over Λ(HL). On the other hand, it is easily seen that our isogeny f induces a Λ(GL)-homomorphism from X(E/F∞) to X(E/F∞), whose kernel is killed by a power ofp. Thus X(E/F∞)belongs to MH(G), and the proof of Proposition 5.6 is complete.
We end this paper by conjecturing the existence of a p-adicL-function attached to E over F∞, and formulating a main conjecture which relates this p-adic L-function to X(E/F∞). We refer the reader to [17] for motivation for the definition we have chosen for our p-adic L-function. If q is any prime number, we write Frobq for the Frobenius automorphism of q in Gal(Q¯ q/Qq)/Iq, where, as usual, Iq denotes the in-ertia subgroup. To fix the definition of our local and global ε-factors, we follow the conventions and normalizations of [13]. In particular, we choose the sign of the local reciprocity map so that q is mapped to Frob−q1. We take the Haar measure on the
ad`ele group of Q which is the usual Haar measure on R, and, for each prime num-berq, gives Zq volume 1. We also fix the unique complex character of the ad`ele group of Q whose infinite component is x → exp(2πix), and whose component at a finite primeq is x→exp(−2πix). Now let ρ denote any Artin representation of G. In our earlier p-adic theory, we viewed ρ as being realized over the ring of integers of some finite extension of Qp. However, since ρ is an Artin representation of G, finite group theory tells us that there exists a finite extension of Q , which we denote by Kρ, such thatρ can be realized in a finite dimensional vector space Vρ over Kρ. We first recall the definition of the complex ArtinL-functionL(ρ,s). It is given by the Euler product
L(ρ,s)=
q
Pq(ρ,q−s)−1,
wherePq(ρ,T) is the polynomial Pq(ρ,T)=det
1−Frob−q1.TVρIq .
For each prime number q, we write eq(ρ) for the local ε-factor of ρ at q, normalized as in [13].
For each prime number l, let Tl(E)=lim
←−Eln, Vl(E)=Tl(E)⊗Zl Ql, H1l(E)=Hom(Vl(E),Ql).
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Moreover, we fix some prime λ of Kρ above l, and put Vρ,λ=Vρ⊗KKρ,λ. The com-plex L-function which is obtained by twisting E by the Artin representation ρ is de-fined as follows.
L(E, ρ,s)=
q
Pq(E, ρ,q−s)−1, (102)
wherePq(E, ρ,T) is the polynomial Pq(E, ρ,T)=det
1−Frob−q1.TH1l(E)⊗Ql Vρ,λIq
;
here l is any prime number different from q. The Euler product L(E, ρ,s) converges only for (s) > 3/2, and the only thing known about its analytic continuation at present is that it has a meromorphic continuation when ρ factors through a soluble extension ofQ . We will assume the analytic continuation of L(E, ρ,s) tos =1 for all Artin characters ρ of G in what follows. The point s = 1 is critical for L(E, ρ,s) for all Artin ρ, in the sense of [14] and the period conjecture of [14] asserts the following in this case. Fix a global minimal Weierstrass equation for Eover Z, and let ω denote the N´eron differential of this equation. Let γ+ (resp. γ−) denote a generator for the
subspace of H1(E(C),Z) fixed by complex conjugation (resp. the subspace on which complex conjugation acts by−1). We define
Ω+(E)=
γ+ω, Ω−(E)=
γ−ω.
Moreover, let d+(ρ) denote the dimension of the subspace of Vρ on which complex conjugation acts by +1, and d−(ρ) the dimension of the subspace on which complex conjugation acts by −1. According to the period conjecture of [14], we have
L(E, ρ,1)
Ω+(E)d+(ρ)Ω−(E)d−(ρ) ∈ ¯Q (103)
for all Artin representations ρ of G. We shall assume (103) in order to define our conjecturalp-adicL-function. We also tacitly suppose we have fixed embeddings of Q¯ into both C and Q¯ p.
Let jE denote the j-invariant of E, and let R denote the set consisting of the primep and all prime numbers q with ordq(jE)< 0. We define
LR(E, ρ,s)=
q/∈R
Pq(E, ρ,q−s)−1. (104)
Finally, we put
pfρ =p-part of the conductor of ρ. (105)
Also, since E is ordinary at p, we have
1−apX+pX2=(1−uX)(1−wX), u∈Z×p; (106)
here, as usual,p+1−ap=(E˜p(Fp)), where E˜p denotes the reduction of E modulop.
It is an important fact, emerging from the study of the Tamagawa number con-jecture (see [17], Theorem 4.2.26), that we have to enlarge the coefficient ring of the Iwasawa algebra in order to define our p-adic L-functions. We briefly explain the mi-nor modifications of the algebraic results of § 2 and § 3 needed. Let A denote the ring of integers of any fixed finite extension ofQp. We define
ΛA(G)=lim←−
U
A[G/U], ΛA(H)=lim←−
W
A[H/W],
whereU (resp.W) runs over the set of all open normal subgroups of G (resp. H). We now defineS(A)to be the set of allf inΛA(G)such thatΛA(G)/ΛA(G)f is a finitely generatedΛA(H)-module. Then the arguments of § 2 generalize immediately to show that S(A) is a left and right Ore set of non-zero divisors in ΛA(G). Again, we let
S(A)∗= ∪n≥0pnS(A). In fact, since A is a freeZp-module of finite rank, one sees easily that
ΛA(G)=Λ(G)⊗ZpA, ΛA(G)S(A)∗ =Λ(G)S∗⊗ZpA.
One also verifies immediately that the theory of evaluation given at the beginning of
§ 3 extends without difficulty to elements of K1(ΛA(G)S(A)∗).
To define our conjectural p-adic L-function, we will take A to be the ring of integers of a finite unramified extension of Qp as follows. Let K(F∞) be the maximal abelian extension of Q , which is contained in F∞, and in which p does not ramify.
Since the maximal abelian extension ofQ contained in F∞ has Galois group which is a p-adic Lie group of dimension 1 and contains Q(µp∞), it is clear thatK(F∞) must be a finite extension ofQ . We recall that ρˆ denotes the contragredient representation ofρ.
Conjecture 5.7. — Assume that p ≥ 5 and that E has good ordinary reduction at p. Let A be the ring of integers of the completion at any of the primes above p of the field K(F∞) defined above. Then there exists LE in K1(ΛA(G)S(A)∗) such that, for all Artin representations ρ of G, we have LE(ρ)= ∞, and
LE(ρ)= LR(E, ρ,1)
Ω+(E)d+(ρ)Ω−(E)d−(ρ) ·ep(ρ)· Pp(ρ,ˆ u−1) Pp(ρ,w−1)·u−fρ; (107)
here ep(ρ) denotes the local ε-factors at p attached to ρ. With A as defined in Conjecture 5.7, let
i:K1(Λ(G)S∗)→K1(ΛA(G)S(A)∗)
be the homomorphism induced by the canonical ring extensionΛ(G)S∗→ΛA(G)S(A)∗. Conjecture 5.8 (The main conjecture). — Assume that p ≥ 5, E has good ordinary reduc-tion at p, andX(E/F∞)belongs to MH(G). Let ξE denote a characteristic element forX(E/F∞). Granted Conjecture 5.7, the images ofi(ξE)andLE inK1(ΛA(G)S(A)∗)/(image of K1(ΛA(G))) coincide.
It is clear that such a main conjecture, when proven, will have many deep con-sequences for the arithmetic of E over F∞ and its subfields. The compatibility of our main conjecture with the Tamagawa number conjecture is already discussed in [17].
We only mention now the following almost immediate corollaries.
Corollary 5.9. — Assume Conjecture 5.8. Then, for each Artin representation ρ of G, χ(G,twρ(X(E/F∞))) is finite if and only if L(E,ρ,ˆ 1)=0.
Corollary 5.10. — Assume Conjecture 5.8. Letρ be an Artin representation of Gsuch that L(E,ρ,ˆ 1) = 0. As in Theorem 3.6, let mρ be the degree of the quotient field of O over Qp, whereρ is given by (12). Thenχ(G,twρˆ(X(E/F∞)))is equal to the mρ-th power of the inverse of the p-adic valuation of the right hand side of (107).
To prove Corollary 5.9, we observe that when X(E/F∞) belongs to MH(G), Lemmma 5.3 and Theorem 3.8 show thatξE(ρ)=0 if and only ifχ(G,twρˆ(X(E/F∞))) is finite. Moreover, for any η in the image of K1(ΛA(G)) in K1(ΛA(G)S(A)∗), we have
|η(ρ)|p = 1 for every continuous representation ρ of G of the form (12). Assuming Conjecture 5.8, Corollary 5.9 follows immediately since LE(ρ) = 0 if and only if L(E, ρ,1) = 0. Similarly, Corollary 5.10 is then clear from formulae (36) and (107).
It also seems worthwhile to point out the following integrality properties of LE and ξE which are implicit in our conjectures. If X(E/F∞) belongs to MH(G), Lemma 5.3 and Theorem 3.8 show that ξE(ρ)= ∞ for every Artin representation ρ of G. Thus Case 1 of Conjecture 4.8 would then imply thatξE belongs to the image of the natural map
Λ(G) 1
p ×
∩Λ(G)S∗ →Λ(G)S∗× →K1(Λ(G)S∗).
On the other hand, the conjectured holomorphy of L(E, ρ,s) at s = 1, gives, via Conjecture 5.7, the assertion that LE(ρ) = ∞ for every Artin representation of ρ ofG, suggesting an analogous integrality property should hold for LE.
Example. — We now illustrate the above conjectures by considering the curve E=X1(11)and p=5, which was already discussed at the end of § 3. In this case, it can be shown thatK(F∞)=Q(µ11)+, the maximal real subfield of the field generated over Q by the 11-th roots of unity. The prime 5 is inert in K(F∞), so that A is the ring of integers of the unique unramified extension of Q5 of degree 5.
The following data is calculated in [15]. We have d+(ρi)=d−(ρi)=2 (i =1,2)
(108)
fρ1 =3, fρ2 =5 (109)
P5(ρ1,X)=1−X, P5(ρ2,X)=1 (110)
P5(E, ρ1,X)=1−X+5X2, P5(E, ρ2,X)=1 (111)
P11(E, ρ1,X)=1, P11(E, ρ2,X)=1 (112)
|e5(ρ1)|5=5−3/2, |e5(ρ2)|5 =5−5/2. (113)
Moreover, (E˜(F5))=5, and so
1−a5X+5X2=1−X+5X2. (114)
Finally, even thoughPq(E, ρi,X) (i =1,2) has degree 8 for all q=5,11, the remark-able calculations of [15] show that
L(E, ρ1,1)
(Ω+(E)Ω−(E))2 = −22 112√
5, L(E, ρ2,1)
(Ω+(E)Ω−(E))2 = −26 1125√
5. (115)
Thus, since both of these L-values are non-zero, Proposition 3.11 shows that Corol-lary 5.9 holds for ρ1 andρ2. Finally, one can use the above data to calculate the right hand side of (107), up to a 5-adic unit, for ρ1 and ρ2. Using (74) of Proposition 3.11, and recalling that ρ1 and ρ2 can both be realized over Z5, we deduce that Corol-lary 5.10 is also valid for ρ1 and ρ2.
We end this paper by confessing that the evidence in support of Conjecture 5.8 is still very modest. WhenE admits complex multiplication and G is still taken to be the Galois group ofQ(Ep∞)overQ , Conjecture 5.8 is true providedX(E/F∞)belongs to the category MH(G) (see Corollary 5.5), since it can be deduced without essential difficulties from the classical two variable main conjecture for elliptic curves with com-plex multiplication, which is proven by the work of Rubin [27] and Yager [40]. Note that, in this case, R just consists of p, since E has integral j-invariant. We would also like to mention that the first numerical evidence in support of the existence of non-abelian p-adic L-functions was found by Balister [3]. In another direction, [15] pro-vides striking and much more extensive numerical evidence for the analogues of Con-jecture 5.7 and Corollary 5.10 forE over the variable p-adic Lie extension of Q given by
Q(µp∞, √pn
m:n =1,2, ...),
where m is some fixed integer > 1. The only change in formulating Conjecture 5.8 for thesep-adic Lie extensions is to now take the set R of primes whose Euler factors are omitted in (104) to consist of p and all primes dividing m. Moreover, in this case, p can be any odd prime where E has good ordinary reduction.
Finally, when G is non-commutative, Conjectures 5.7 and 5.8, as well as Con-jecture 4.8, suggest that it is important to find an explicit description of theK1 of the Iwasawa algebra of G and its localization at the set S∗ (see [21] for the case when G is the semidirect product of Zp with Z×p).
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