A L
E S
E D L ’IN IT ST T U
F O U R
ANNALES
DE
L’INSTITUT FOURIER
Matteo LONGO
On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields
Tome 56, no3 (2006), p. 689-733.
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ON THE BIRCH AND SWINNERTON-DYER CONJECTURE FOR MODULAR ELLIPTIC CURVES
OVER TOTALLY REAL FIELDS
by Matteo LONGO (*)
Abstract. — LetE/F be a modular elliptic curve defined over a totally real number fieldF and letφbe its associated eigenform. This article presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of Eover suitable quadratic imaginary extensionsK/F. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when[F:Q]is even andφnot new at any prime.
Résumé. — SoitE/F une courbe elliptique modulaire définie sur un corps de nombres totalement réelF et soitφla forme propre associée. Cet article présente un nouvelle méthode, inspirée par un récent travail de Bertolini et Darmon, pour contrôler le rang de E sur des extensions convenables quadratiques imaginaires K/F. En particulier, ce résultat peut être appliqué aux cas qui ne sont pas consi- dérés dans le travail de Kolyvagin et Logachëv, i.e.,quand[F :Q]est pair etφ n’est pas nouveau en aucun idéal premier.
Introduction
LetEbe an elliptic curve of conductornwith no complex multiplication defined over a totally real number fieldF of finite degreedoverQ. Choose a totally imaginary quadratic extension K/F; such an extension can be described as K = F(√
α), where α is a totally negative element of F. By the well-known theorem of Mordell-Weil, the rank of the groupsE(F) andE(K)are finite. Denote by L(E, s)the Hasse-Weil L-series attached
Keywords:Elliptic Curves, Birch and Swinnerton-Dyer Conjecture, Shimura Varieties, Congruences between Hilbert Modular Forms.
Math. classification:11G05, 11G18, 11G40, 11F30.
(*) The author was partially supported by the research project COFIN PRIN 2002 Geometria delle Varietà Algebricheand by the Marie Curie Research Training Network Arithmetic Algebraic Geometry.
to E and by LK(E, s) its base change over K. These series converge for
<(s)greater that3/2. The following Birch and Swinnerton-Dyer conjecture BSDforE overF andK is well-known:
Conjecture (BSD). — The series L(E, s) and LK(E, s) have a con- tinuation to entire functions whose order of vanishing at the central point s= 1is equal to the rank of the groups, respectively,E(F)andE(K).
This article presents somenewcases of theBSDconjecture formodular elliptic curves E/F when the order of vanishing of LK(E, s) at s = 1 is zero. These new cases are those not covered by the well-known work of Kolyvagin and Logachëv [34]. In particular, the present work can be applied to modular elliptic curvesE with everywhere good reduction over extensions F/Q of even degree (and to all the twists of E by quadratic characters ofF).
The notion of modularity can be made precise as follows. For any rational primep, denote byTp(E)thep-adic Tate module ofEand by
ρE,p: Gal(F /F)→Aut(Tp(E))'GL2(Zp) the associated Galois representation.
On the other hand, let φ∈ S2(n) be a Hilbert modular form of parallel weight 2 andΓ0(n)-level structure (see Section 1.1 for precise definitions).
Assume thatφis an eigenform for the action of the Hecke algebraTnacting faithfully onS2(n)and denote by θφ(T) the associated eigenvalues, where T∈Tn. For any prime idealq-n(respectively,q|n) ofOF, denote byTq
the Hecke operator atqand bySqthe spherical operator atq(respectively, by Uq the Hecke operator at q); see [42] for definitions. After a suitable normalization, it is possible to assume that the eigenvalues θφ(T) belong to the ring of integersOφ of a finite extensionKφofQ. Fix finally a prime idealp of Oφ. Thanks to the work of Carayol [11], Wiles [50] and Taylor [45], there is a unique continuous representation:
ρφ,p: Gal(F /F)→GL2(Oφ,p)
which is unramified at all the prime idealsq-npand so that the character- istic polynomial of a Frobenius element at these primes isX2−θφ(Tq) +
|q|θφ(Sq), whereOφ,p is the completion ofOφ at p and |q| is the norm of q, that is, the number of elements of the residue field ofOF atq.
The following definition explains the notion of modularity.
Definition 0.1. — E ismodular if there exists an eigenformφof par- allel weight 2 and leveln so thatKφ =Qand ρφ,p is equivalent to ρE,p, wherepis a prime ofZ.
IfEis modular, thenL(E, s)(respectively,LK(E, s)) coincides with the L-seriesL(φ, s)(respectively, LK(φ, s)) for<(s)0. Hence, L(E, s) and LK(E, s)have continuations to entire functions. In the following, we prefer the notationsL(φ, s)andLK(φ, s)to denote L(E, s)andLK(E, s).
ForF =QtheBSDconjecture is known when the order of vanishing of L(E, s) =L(φ, s)ats= 1is at most one. Thanks to the work of Wiles [51]
and Taylor-Wiles [46], successively improved in a series of papers [17], [14]
and [8], it is known that all elliptic curves overQare modular. SetN :=n.
For such curves, there is a parametrization overQ: ϕ:X0(N)→E,
whereX0(N)is the modular curve of levelN. Letωbe the unique invariant differential onE overQso thatϕ∗(ω)(z) =φ(z)dz(za complex variable).
Suppose that all primes dividing N are split in K := Q(√
−D). In this case, the order of vanishing of LK(φ,1) is odd, hence LK(φ,1) = 0. The BSDconjecture for elliptic curves over K should imply that the rank of E(K)is at least one, and exactly one ifL0K(φ,1)6= 0.
There is a theory of Heegner points onX0(N)coming from its interpre- tation as moduli space for elliptic curves with a cyclic subgroup of orderN. More precisely, choose an idealN ofOK, the ring of integers ofK, so that OK/N 'Z/NZ. The complex toriC/OK andC/N−1define elliptic curves related by a cyclicN-isogeny, giving a complex pointx1∈X0(N)(C). The theory of complex multiplication implies that this point is defined over the Hilbert class fieldK1ofK. Definey1:=ϕ(x1)andyK := TrK1/K(y1). The main result of [25] is:
L0K(φ,1) = R
E(C)ω∧iω
√D
ˆh(yK),
whereˆhis the Néron-Tate height. It follows that the point yK has infinite order if and only ifL0K(φ,1)6= 0.
In [33] Kolyvagin proved that, under the previous assumptions, the rank ofE(K)is one and that
yK∈E(K)υ,
where−υ=±1is the sign of the functional equation ofL(φ, s)andE(K)υ is theυ-eigenspace for the complex conjugation. For more details and an exposition of this argument, see [24].
Kolyvagin’s result proves the rank one case of the BSD conjecture for Eover Kand can be used to derive a proof of the Birch and Swinnerton- Dyer conjecture for E over Q when the order of vanishing of L(φ, s) at s= 1is at most one. More precisely, assume that the order of vanishing of
L(φ,1)is 0 or 1. In this case, it is possible to choose an extensionK/F so that all primes dividingN are split inK andL0K(φ,1)6= 0(ifL(φ,1)6= 0 the existence of such a fieldK follows from the work of Bump-Friedberg- Hoffstein [9] and Murty-Murty [37], while forL(φ,1) = 0this is a result of Waldspurger [48]). Then Kolyvagin’s result on the rank ofE(K)imply the BSDconjecture overQ.
Assume now that either [F : Q] is odd or [F : Q] is even and φ is new at least at one prime which dividesnexactly. In this case the previous techniques can be generalized. The main idea is to replace, via the Jacquet- Langlands correspondence, the modular parametrization by a Shimura cur- ve parametrization ϕ : X → E defined over F in the spirit of [11]. The analogue in this context of [25] and [33] are, respectively, [53] and [34]. See [53] for more details.
This article proposes a new approach to the BSD conjecture when LK(φ,1)6= 0. A similar strategy has been used by Bertolini and Darmon [4] in the context of Iwasawa’s Main Conjecture. It is worth to point out that the totally real case introduces new problems which are not present in the rational case and must be treated by different tools. The main features of this approach are:
1. The analytic result on the non-vanishing of L0K(φ,1) is replaced by a more algebraic point of view such as the Gross formula [23]
generalized by Zhang in [52].
2. Kolyvagin’s method requires an imaginary quadratic K/F so that the rank ofE(K)is one, while in the setting of this work the rank ofE(K)is zero.
3. It is possible to treat the missing cases for F totally real, that is, [F : Q] even and φ not new at any prime. In particular, this method applies to elliptic curves with everywhere good reduction defined over totally real fields of even degree overQand to all their twists by quadratic characters ofF. This case introduces arithmetic problems which are not present in the caseF =Q.
The main idea of this approach is parallel to the idea that [50] and [45]
used to build thep-adic representation associated to a modular formφwhen [F :Q]is even andφis not new at any prime. Basically, by the theory of congruences between modular forms, it is possible to find a modular form φ`≡φ (modp)of leveln`which is new at`, where`⊂ OF is a congruence prime. Sinceφis new at`, there is a theory of Shimura curves associated toφ`and, by varying the prime`, it is possible to obtain an Euler system and control the rank ofE(K).
The main result
Although the most intriguing aspects of this work concern elliptic curves overF with everywhere good reduction and[F :Q] even, the main result and the outline of the proof are now stated in a more general form.
If the relative discriminant of K/F is prime ton, there is a factorization n=n+n− induced by the extensionK/F:n+(respectively,n−) is divisible by the prime ideals dividingnwhich are split (respectively, inert) inK.
Assumption 0.2. — The idealnsatisfies the following conditions:
1. The discriminant ofK/F is prime to n; letn=n+n− be the asso- ciated factorization as above;
2. n− is square-free and the number of primes dividing it has the same parity asd:= [F :Q];
3. φis new at each prime dividingn−.
IfE has everywhere good reduction and[F :Q]is even, Assumption 0.2 is verified for any K. This is true also for any twist of E by quadratic charactersχofF such that the corresponding quadratic extension ofF is eitherCMor totally real. This assumption is equivalent to the requirement that the order of vanishing ofLK(φ,1) is even, so it is compatible with the hypothesis LK(φ,1) 6= 0. As already observed, the BSD conjecture precludes in this setting the existence of Heegner points with infinite order as in the original method of Kolyvagin. Moreover, the parity assumption is crucial for the use of Shimura curves.
The following Theorem A (respectively, Theorem B) states that the BSDconjecture holds when the order of vanishing ofL(φ, s)(respectively, LK(φ, s)) ats= 1is zero.
Theorem A. — IfL(φ,1)6= 0 thenE(F)is finite.
Theorem B. — IfLK(φ,1)6= 0thenE(K)is finite.
Theorem A follows from Theorem B by non-vanishing results for twists of L-series. Indeed, ifL(φ,1)6= 0then by [49] it is possible to findKinducing a factorizationn=n+n− as above and so that LK(φ,1)6= 0.
Theorem B follows by making ap-descent for a suitable rational primep and bounding thep-Selmer groupSelp(E/K)(the definition ofSelp(E/K) can be found in [44, Ch. X, §4]). The next paragraphs explain the conditions which are required for the choice ofp. Theorem B will be deduced from Theorem C in the following after the choice of a suitable primep.
TheGross-Zhang formula[52] gives an arithmetic description of the spe- cial value at s = 1 of LK(φ, s). Let B be a quaternion algebra over F
which is ramified at all the archimedean places ofF and at all the prime ideals dividingn− (by the previous assumption, such a quaternion algebra exists). Let R ⊆ B be an Eichler order of level n+. Denote by Bb and Rb the finite adele ring of, respectively,B and R. By the Jacquet-Langlands correspondence [27] there exists a weight 2 modular form
f :Rb×\Bb×/B×→Z
on the quaternion algebra B with the same eigenvalues as φ under the action of the Hecke algebra Tn. Since any prime dividing n− is inert in K/F by Assumption 0.2, it follows from [47, Chapitre III, Théorème 3.8]
that there exists an embeddingΨ :K ,→B so that Ψ(OK) = R∩Ψ(K), whereOK is the ring of algebraic integers ofK. Then there is a map:
Ψ :b ObK×\Kb×/K× →Rb×\Bb×/B×. Define the algebraic part ofLK(φ, s)to be: LK(φ) :=P
a(f ◦Ψ)(a)b ∈Z, where the sum is over a set of representatives ofPic(OK)'ObK×\Kb×/K×. The Gross-Zhang formula states that:
(0.1) LK(φ,1)=· |LK(φ)|2,
where=· denotes an equality up to an explicitly computable non-zero factor.
For any group Gdenote byG[p]itsp-torsion. Define Selp(φ/K) :=
s∈Selp(E/K) : resq(s) = 0,∀q|n+ ,
where resq(s) : H1(K, E[p]) → H1(Kq, E[p]) := ⊕v|qH1(Kv, E[p]) is the direct sum of the restriction maps in Galois cohomology, the sum is over the set of primesv of OK dividingq andKv is the completion ofK at v (for more details on these definitions, see Section 1.3).
Say that a modular form φ to be p-isolated if there are no non-trivial congruencesφ≡ψ (mod p)betweenφand other formsψof levelnwhich are new atn−.
The following result provides the key ingredient to prove Theorem B above.
Theorem C. — Assume that the following conditions on the prime p >3are verified:
1. pis prime tonD, where Dis the absolute discriminant ofK;
2. p-LK(φ);
3. φisp-isolated;
4. TheGal(F /F)-moduleE[p]is irreducible.
ThenSelp(φ/K) = 0.
Theorem B can be deduced by Theorem C as follows. Assume that LK(φ,1) 6= 0. Choose a rational prime p > 3 verifying conditions 1, 2, 3 and 4 in Theorem C and the following displayed equation:
(0.2) Selp(E/K) = Selp(φ/K).
Note that there are infinitely many primes satisfying all these conditions:
for 2, sinceLK(φ,1)6= 0, Equation (0.1) shows thatLK(φ)6= 0; for 3, use the finiteness of theC-vector space of Hilbert modular forms of fixed weight and level; for 4, use [41, Théorème 2] and the fact thatE has no complex multiplication; for the displayed Equation (0.2), choosepsuch thatE has good ordinary reduction at primes dividingpand use the theorem of Lutz as in [22, Section 2] (recall thatpis prime ton+ by 1). By the choice ofp made above, it follows from Theorem C thatSelp(E/K) = 0. Since there is an injective map
E(K)/pE(K),→Selp(E/K) arising from Kummer theory, Theorem B follows.
The proof of Theorem A and B is then reduced to the proof of Theorem C, which is outlined in the next subsection.
The outline of the proof of Theorem C
The general approach for obtaining results on the rank of E(K) is to bound thep-Selmer groupSelp(E/K). The strategy for finding such bounds is to construct a collection of global cohomology classes (a so calledEuler system)
κ`∈H1(K, E[p])
`∈L so that: (1)L is a sufficiently large set of primes ofOFand (2) each classκ`satisfies prescribed local properties, that is, the restriction ofκ`at any prime not dividingn+`belongs to the image of the local Kummer map. The existence of an Euler system combined with a standard argument based on the global reciprocity law of class field theory can then be used to obtain the desired bound.
The idea of the present paper, as in [4], is to produce an Euler system using the theory of congruences between modular forms. The set of primes Lfor the Euler system{κ`}`∈Lis given by the congruence primes`, which, following the terminology introduced by Bertolini and Darmon, are called p-admissible. For any`∈L, the class κ` is obtained from Heegner points on the Shimura curveXn+,n−` of leveln+ attached to a quaternion algebra of discriminantn−`which is split at exactly one of the archimedean places ofF. The sketch of the proof can be divided into five steps.
Step 1: The raising the level result.
Let θφ :Tn →Z/pZ denote the morphism associated to φ. A modular form φ0 of level n0 is said to be congruent to φ (modp) (write: φ0 ≡ φ (modp)) if the Fourier coefficients of φ0 belong to the ring of integer Oφ0
of a number field Kφ0 and there is a prime ideal p ⊆ Oφ0 dividing p so that: (1) Oφ0/p ' Z/pZ and (2) Tq(φ0) ≡ θφ(Tq)φ0 (modp) for primes q-nn0 andTq(φ0)≡θφ(Uq)φ0 (modp)for primesq|(n,n0). Definition 0.3 introduces the congruence primes and Theorem 0.4 states the raising the level result.
Definition 0.3. — Define a prime`⊂ OF to be p-admissible if:
1. `-np;
2. `is inert inK;
3. p-|`|2−1;
4. p| |`|+ 1−θφ(T`), where =±1.
Theorem 0.4. — Assume that`isp-admissible and that the represen- tation of Gal(F /F) on E[p] is irreducible. Then there exists a modular formφ` of leveln` new at`so thatφ`≡φ (modp).
Step 2: The Jacquet-Langlands correspondence.
Denote by θφ : Tn → Z/pZ (respectively, θφ` : Tn` → Z/pZ) the mor- phism associated to φ (respectively, to φ`). Use the following notations:
mf := Ker(θφ)andmf` := Ker(θφ`). By the Jacquet-Langlands correspon- dence it is possible to associate toφ` a modular form f` on a quaternion algebraBwhich is split at exactly one of the archimedean places ofF and whose discriminant isn−` (note that such a quaternion algebra exists by Assumption 0.2). Fix an Eichler orderR ⊆ Bof leveln+ and consider the Shimura curveX defined overF whose complex points are
X(C) :=Rb×\(Bb×× H±)/B×'
h
a
j=1
Γj\H,
whereH± =C−R, the symbol Hdenotes the complex upper half plane and Γj ⊆ B× are arithmetic subgroups related to the level structure R.
The modular formf`can be viewed as a holomorphic differential onX(C).
Denote by J the Jacobian variety of X and by J[p] its p-torsion. The following theorem, due essentially to Boston-Lenstra-Ribet [6], gives the relation between the cohomology ofE[p]and that of J[p].
Theorem 0.5. — There is an isomorphism of Gal(F /F)-modules J[p]/mf` 'E[p]k,wherekis a positive integer.
Step 3: The construction ofκ`.
Denote by µthe archimedean place of F where B is split. By Assump- tion 0.2 there is an embedding
K×,→ B× ,→(B ⊗F,µR)×'GL2(R)
inducing an action of K× on H± by fractional linear transformations.
This action has only one fixed point P ∈ H which is rational over the Hilbert class fieldK1ofKby the theory of complex multiplication (see [43, Theorem 9.6]). Define the Heegner divisor DK := P
σ∈Gal(K1/K)σ(P) ∈ Div(X)(K1). If the representation ofGal(F /F)onE[p]is irreducible, the idealmf` is not Eisenstein and soDK defines a point inPK ∈J(K)/mf`. The Kummer map:
(0.3) J[p]/mf` ,→H1(K, J[p]/mf`)'H1(K, E[p])k
yieldskglobal classesκj∈H1(K, E[p])forj = 1, . . . , k (the isomorphism in (0.3) is a consequence of Theorem 0.5). For any primeq⊆ OF, denote by δq the local Kummer map and byresq the restriction map in cohomology.
Since anyκjcomes from a class ofSelp(J/K)and the conductors ofJ and E differ only at`, it is possible to show thatresq(κ`)∈Im(δq)for primes q-n+`(see Theorem 5.5 for precise references). So there are two problems:
(1) The choice of a suitable component H1(K, E[p]) ⊆ H1(K, J[p]/mf`):
the classκ` is then defined to be the projection on it; (2) The description ofres`(κ`).
Remark 0.6. — The problem of finding a suitable copy ofH1(K, E[p]) inH1(K, J[p]/mf`)appears only when[F :Q]is even andφis not new at any prime dividingn−. Indeed, in all other cases (so, in particular, when F=Q) it is possible to show thatΦ`/mf` 'Z/pZand, as a consequence, to prove thatJ[p]/mf` 'E[p]. The problems with the missing case are related to the geometry of Shimura curves and to the description of the action of Hecke operators via Brandt matrices. More precisely, assume from now to the end of this remark that[F :Q]is even and φis not new at any prime dividing n−. In this case, it is not possible prove that Φ`/mf` ' Z/pZ using the the argument of [38, Proposition 5] based on the geometry of Shimura curves. On the other hand, it is possible, studying the action of Hecke operators via Brandt matrices, to give precise conditions onF andn
which implyΦ`/mf` 'Z/pZ. These conditions are collected in the notion ofEisenstein pair (see Definition 3.4). When no one of these conditions is verified, it is necessary to use a more complicate argument to find a suitable component C` 'Z/pZ ⊆Φ`/mf` playing the same role as Φ`/mf` in the previous case.
Step 4: The componentC` and the Reciprocity Law.
SinceX is a moduli space for suitable abelian varieties with level struc- ture, there is a modelX of X over OF. Denote by X` the fiber ofX at
`and defineX`2 :=X`⊗F` K`. Let C` be the completion of the algebraic closure ofF` and defineH`:=C`−F`. By the Čerednik-Drinfeld theorem, there is an isomorphism of rigid analytic spaces overK`:
X`2(C`) =
h
a
j=1
Γj\H`,
whereΓj ⊆PGL2(F`) are arithmetic subgroups related to the fixed level structureR ⊆ B. Denote byXF
`2 the special fiber ofX`2, whereF`2 is the residue field ofK`. The Čerednik-Drinfeld theorem shows that the set of verticesV`of the arithmetic graph G`associated to XF
`2 has the following description:V`=Rb×\Bb×/B×× {0,1}.
Theorem 0.7. — The Heegner point P ∈ X(K1) reduces to a non singular point of the special fiberXF
`2. In particular,PK defines a divisor vK∈Z0[Rb×\Bb×/B×]/mf`.
This result is proved in Section 5.2 using the`-adic description of Heegner points obtained from the Čerednik-Drinfeld theorem.
LetΦ`denote the group of connected components of the Jacobian variety of X`2. Define Hsing1 (K`, E[p]) := H1(K`unr, E[p])Gal(K`unr/K`). This group is the orthogonal complement of δ`(E(K`)/pE(K`))under the local Tate pairing h , i`. The following Theorem 0.8 (proved in Propositions 4.10 and 4.11) characterizes the choice of the component H1(K, E[p]) inside H1(K, J[p]/mf`), while Theorem 0.9 (proved in Proposition 5.4) describes res`(κ`)by the Reciprocity Law.
Theorem 0.8. — There exists a component C` ' Z/pZ ⊂ Φ`/mf` so that
C`'Hsing1 (K`, E[p])⊂H1(K`, E[p])⊂H1(K`, J[p]/mf`)
and there is a canonical non-trivial isomorphism
ω`: Im(δ∗)/hmf,U2`−1i → C`'Z/pZ,
whereIm(δ∗)⊆Z0[Rb×\Bb×/B×]is the subgroup of elements having degree zero on each connected component ofG`.
Theorem 0.9. — There is an elementC ∈(Z/pZ)× so thatω`(vK)≡ CLK(φ) (modp)inC`.
Step 5: The Euler system argument.
This is the final step of the proof of Theorem C. We must show that Selp(φ/K) is zero. So, fix an element s ∈ Selp(φ/K) and assume that s 6= 0. For any primes q, let ∂q : H1(K, E[p]) → Hsing1 (Kq, E[p]) be the composition of the mapresqand the projection toHsing1 (Kq, E[p]). Choose a p-admissible prime`so that∂`(s) = 0andres`(s)6= 0(such a prime`exists by Theorem 2.3). If` is ap-admissible prime, then (see also Lemma 2.2) (0.4) δ`(E(K`)/pE(K`))'Z/pZ and Hsing1 (K`, E[p])'Z/pZ. The global Tate duality yieldshs, κ`i=P
qhresq(s),resq(κ`)iq = 0. Since resq(κ`) is orthogonal to s with respect to the local Tate pairing h , iq forq 6=`, it follows thathres`(s),res`(κ`)i` = 0.Since Hfin1 (K`, E[p]) and Hsing1 (K`, E[p])are orthogonal with respect to the local pairingh, i`, the isomorphisms (0.4) and the condition∂`(κ`)6= 0(which follows from The- orem 0.9) imply thatres`(s) = 0, which is a contradiction. This completes the sketch of the proof.
Examples.
Assume thatE is an elliptic curve with everywhere good reduction over a real quadratic fieldF =Q(√
D). IfEis aQ-curve, that is,Eis isogenous to its Galois conjugate, thenE appears as a quotient of the modular curve J1(N)overQfor someN. In this case, the classical methods of Kolyvagin could perhaps be applied toE (see the brief discussion in [16, Section 3]).
The author will study this variant of the classical method in a forthcoming work. The really new cases which can be treated by the present work are elliptic curves which are notQ-curves. An example of such curves can be found in [16]: let ω := 1+
√509
2 and F = Q(ω); then the elliptic curve corresponding to the following Weierstrass equation
y2−xy−ωy=x3+ (2 + 2ω)x2+ (162 + 3ω)x+ (71 + 34ω)
is not aQ-curve. For this curve, and also for any of its twists by quadratic characters ofF, the present result is really new and could not be obtained by other methods.
Acknowledgements. The results contained in this article are presented in the author’s Ph.D. thesis [35]. It is a pleasure for the author to thank sincerely his Ph.D. advisor Professor Massimo Bertolini for having proposed this problem and for very helpful suggestions and improvements during the work. The author wishes also to thank Professor Frances Sullivan for reading the preliminary version of this paper and the anonymous referee for suggesting useful improvements in the exposition.
1. Selmer groups of modular elliptic curves
1.1. Hilbert modular forms
Let F be a totally real number field of finite degreedoverQwith ring of algebraic integers OF. For any open compact subgroup U ⊆GLc2(F), denote byS2(U)the finite dimensionalC-vector space of parallel weight 2 Hilbert cusp forms with respect toU (hereGLc2(F)is the idele group of the ringM2(F)of2×2matrices with coefficients inF). For the definition and the main properties of this space, see [45] and [50]. SetS2(n) :=S2(bΓ0(n)), whereΓ0(n)is the subgroup ofM2(OF) consisting of (mod n)upper tri- angular matrices andΓb0(n)denotes its idele group. Finally, for any divisor r|n, letS2new(n/r,r)⊆ S2(n)be the subspace of those forms which are new atrand set:S2new(n) :=S2new(OF,n).
Let Tn be the Hecke algebra of level n acting faithfully on S2(n). For prime idealsq-n(respectively,q|n) denote byTq(respectively, byUq) the corresponding Hecke operator. Finally, for primesq-n, denote bySq∈Tn
the spherical operator atq. For the definitions of these operators, see [42].
Denote byTnewn/r,r the quotient of Tn acting faithfully on S2new(n/r,r) and setTnewn :=TnewOF,n.
Anyφ∈ S2(n)which is a simultaneous eigenvector for all Hecke operators gives rise to a morphism
θφ:Tn→C
so that, for anyT∈Tn,T(φ) =θφ(T)φ.An eigenvectorφas above can be normalized so that the image ofθφ is an order in the ring of integers Oφ of a finite extensionKφ ofQ; such aφis called a normalized eigenform or
simply an eigenform. For any primepofOφ, denote byOφ,pthe completion ofOφ,p atp and by
θφ:Tn→ Oφ,p/p the reduction ofθφ (mod p).
Let φ ∈ S2(n) be a normalized eigenform and fix a prime ideal p of Oφ. Denote by F an algebraic closure ofF and byOφ,p the completion of Oφ at p. Recall from the Introduction that there is a unique continuous representation of the absolute Galois group ofF:
ρφ,p: Gal(F /F)→GL2(Oφ,p)
which is unramified at all the prime idealsq- np and so that the charac- teristic polynomial of a Frobenius element at these primes is
X2−θφ(Tq) +|q|θφ(Sq).
Assume now thatKφ=Q; then p=pis a rational prime andOφ,p =Zp. In this case, say that φhas rational coefficients. Denote by Tφ[p∞] 'Z2p
theGal(F /F)-module associated to the representationρφ,p and define Vφ[p∞] :=Tφ[p∞]⊗Qp'Q2p.
Set:Tφ[pn] :=Tφ[p∞]/pnTφ[p∞]'(Z/pnZ)2 so that the multiplication by p:Tφ[pn+1]→Tφ[pn]yields a projective system and:Tφ[p∞] = lim←nTφ[pn].
On the other hand, define:
Aφ[p∞] :=Vφ[p∞]/Tφ[p∞]'(Qp/Zp)2
and denote by Aφ[pn] ' (Z/pnZ)2 its pn-torsion. The natural inclusion Aφ[pn],→Aφ[pn+1]yields an inductive system and:Aφ[p∞] = lim→nAφ[pn].
Note that the Galois modulesTφ[pn]andAφ[pn]are isomorphic. Reduction modulopofρφ,p yields a representation:
ρφ,p: Gal(F /F)→GL2(Z/pZ)
whose associatedGal(F /F)-module isAφ[p](or, that is the same,Tφ[p]).
Letφ∈ S2(n)be a normalized eigenform and letθφandρφ,pbe as above.
Denote bymφ,pthe kernel ofθφ. For any finite setSof prime ideals ofOF, denote byT(S)n the subalgebra ofTngenerated byTqandSqforq-n,q6∈S andUq forq|n,q6∈S. Definem(S)φ,p:=mφ,p∩T(S)n .
Definition 1.1. — AT(S)n -moduleEis said to be Eisensteinif its com- pletionEm(S)
φ,p
is zero for any maximal idealm(S)φ,p defined as above and such that the residual representationρφ,p is irreducible.
Remark 1.2. — See also the characterization of Eisenstein modules in- troduced by [18] forF =Qand generalized by [28].
Proposition 1.3. — For any integral ideal q - n of OF, define ηq :=
Tq−(|q|+ 1). Let S be a finite set of prime ideals. If ρφ,p is irreducible, then the ideal hηq,q - n,q 6∈ Si and the maximal ideal mφ,p are prime to each other. It follows that, if Tq ∈ T(S)n acts on a T(S)n -module E as multiplication by|q|+ 1, thenE is Eisenstein.
Proof. — The proof is a direct generalization of [39, Theorem 5.2, part c]
1.2. Modular elliptic curves
Let E/F be an elliptic curve with no complex multiplication defined over the totally real number fieldF. Denote bynits arithmetic conductor, which is an integral ideal ofOF. For any rational primep, denote byE[p]
thep-torsion ofE and byTp(E)itsp-adic Tate module. Denote finally by ρE,p: Gal(F /F)→Aut(Tp(E))'GL2(Zp)
and by
ρE,p: Gal(F /F)→Aut(E[p])'GL2(Z/pZ)
the representation of the absolute Galois group ofF, respectively, onTp(E) andE[p].
Definition 1.4. — The elliptic curve E of conductor n is said to be modularif there exists an eigenformφ∈ S2(n)with rational coefficients so that theGal(F /F)-modulesTp(E)and Tφ[p∞]are isomorphic, wherepis a rational prime.
1.3. The Selmer group
LetE be a modular elliptic curve of conductornand letφbe its associ- ated eigenform. Assume that the primep, appearing in the Definition 1.4, satisfies the following:
Assumption 1.5. — Ehas good ordinary reduction at each primep|p ofOF and the representationTφ[p]is irreducible.
Note that Assumption 1.5 implies in particular that n− and p are rel- atively prime. Moreover, sinceE has no complex multiplication, Assump- tion 1.5 is verified for infinitely many primes by [41] and [5]. Choose a to- tally imaginary quadratic extensionK/F with relative discriminantDK/F prime tonp. The extensionK defines a factorizationn=n+n− where n+ (respectively,n−) is divisible by the prime ideals ofF which are split (re- spectively, inert) inK. The techniques which will be used in this work rely on the following:
Assumption 1.6. — n− is square-free and the number of prime ideals of OF dividing n− and d := [F : Q] have the same parity. Moreover, φ∈ S2new(n+,n−), that is,φis new at the primes dividingn−.
Remark 1.7. — The above conditions are obviously verified if[F :Q]is even andn− =OF, which is the leading example and the most interesting application of this work.
For any field k and any Gal(k/k)-module M, denote by H1(k, M) = H1(Gal(k/k), M) the continuous cohomology groups (here k is an alge- braic closure ofk and i >0). If q is a prime ideal of OF, let Kq be the sum of the completions of K at the primes v | q of the ring of integers OK of K. Choose decomposition subgroups Gq ⊂ Gal(F /F) at q and Gv ⊂Gal(K/K)atv|qso thatGv ⊆Gq. LetIq⊂GqandIv ⊂Gvbe the inertia subgroups. Finally, letKvunrbe the maximal unramified extension of Kv. For anyGal(K/K)-moduleM, set:H1(Kq, M) :=⊕v|qH1(Kv, M)and H1(Iq, M) :=⊕v|qH1(Iv, M),where the sum is extended over all the prime idealsvofK dividingq. For any positive integern, denote byMφ[p]either Aφ[p]orTφ[p]. In view of the definition of thep-Selmer groupSelp(φ/K)as- sociated toφ,pandK, the following notions of finite, singular and ordinary structures are introduced.
Good primes
For primes q-npofOF, define thesingular partofH1(Kq, E[p])to be:
Hsing1 (Kq, Mφ[p]) :=⊕v|qH1(Kvunr, Mφ[p])Gal(Kunrv /Kv). Thefinite partis the kernel of the natural projection map:
Hfin1 (Kq, Mφ[p]) := Ker(H1(Kq, Mφ[p])→Hsing1 (Kq, Mφ[p])).
Primes dividingn−
If q |n−, assume that p-|q|2−1. By the Tate uniformization of ellip- tic curves, there exists an unique subspace Mφ(q)[p] ' Z/pZ of Mφ[p] so that Gal(Fq/Fq) acts on it by±p, where p : Gal(Fq/Fq) → Z×p is the cyclotomic character describing the action on the p-power roots of unity (ifMφ[p] is unramified atq, use that a Frobenius element atq acting on Mφ[p]has eigenvalues±1and±|q|and thatp-|q|2−1: see [4, Remark after Assumption 2.1] for more details). In this case there is an exact sequence ofGal(F /F)-modules:
0→Mφ(q)[p]→Mφ[p]→Mφ(1)[p] :=Mφ[p]/Mφ(q)[p]→0
and Iq acts trivially on the last quotient. Define the ordinary part of H1(Kq, E[p])to be:
Hord1 (Kq, Mφ[p]) :=⊕v|qH1(Kv, Mφ(q)[p]).
Primes dividingp
If p | p, by the good ordinary assumption on the reduction of E at p, there exists a unique quotientMφ(1)[p]'Z/pZofMφ[p]so that the inertia subgroupIpatpacts trivially on it (see for example [15, Proposition 2.11]).
Denote byMφ(p)[p]'Z/pZthe kernel of the natural projection map so that there is an exact sequence ofIp-modules:
0→Mφ(p)[p]→Mφ[p]→Mφ(1)[p]→0
andIp acts onMφ(p)[p]via the cyclotomic character. Theordinary part of H1(Kp, E[p])is defined to be:
Hord1 (Kp, Mφ[p]) := R−1p (H1(Iv, Mφ(p)[p])),
whereRp :⊕v|pH1(Kv, Mφ[p])→ ⊕v|pH1(Iv, Mφ[p])is the restriction map.
The Tate duality
For any positive integer n, the Galois modulesAφ[p] and Tφ[p] are iso- morphic to their own Kummer duals and there is a canonicalGal(F /F)- equivariant pairingAφ[p]×Tφ[p]→µp.Combining this with the cup prod- uct in cohomology yields, for each prime q of OF, a canonical local Tate pairing:
h,iq :H1(Kq, Aφ[p])×H1(Kq, Tφ[p])→Qp/Zp.
Proposition 1.8. — Ifq-npthe groups
Hfin1 (Kq, Aφ[p]) and Hfin1 (Kq, Tφ[p])
are annihilators to each other under the local Tate pairingh,iq. The same is true forHord1 (Kq, Aφ[p])andHord1 (Kq, Tφ[p])for primesq|n−p.
Proof. — This result follows from standard properties of the local Tate
pairing: see for example [15, Section 2.3].
Selmer groups For any prime ideal qofOF, denote by
resq:⊕v|qH1(Kv, Mφ[p])→ ⊕v|qH1(Iv, Mφ[p]) the restriction map. For primesq-np, let
∂q:H1(K, Mφ[p])→Hsing1 (K, Mφ[p])
be the composition of the restriction with the projection. If∂q(s) = 0, then denote byvq(s)the image ofsin Hfin1 (K, Mφ[p]).
Definition 1.9. — The Selmer group Selp(φ/K) is the subgroup of the global cohomology groupH1(K, Aφ[p]) consisting of those elements s so that:
1. Forq-np, ∂q(s) = 0.
2. For primesq|n−p,resq(s)∈Hord1 (Kq, Aφ[p]).
3. For primesq|n+,resq(s) = 0.
2. The Euler system argument 2.1. Admissible primes
Keep the same notations and assumptions as in Section 1:Eis a modular elliptic curve andφis its associated eigenform.
Definition 2.1. — A prime ideal`ofOF is said to be admissible(rel- atively to the primep) if the following conditions hold:
(i) `does not dividenp;
(ii) `is inert inK;
(iii) pdoes non divide|`|2−1;
(iv) pdivides|`|+ 1−a`, where=±1anda`=θφ(T`).
Note that if` is an admissible prime thenK`:=K⊗FF`'F`2,where F`2 is the unique quadratic unramified extension ofF`.
Lemma 2.2. — Let`be an admissible prime. Then the local cohomology groupsHfin1 (K`, Aφ[p])andHsing1 (K`, Tφ[p])are both isomorphic toZ/pZ.
Proof. — A direct generalization of [4, Lemma 2.6].
Theorem 2.3. — Let s be a non-zero element of H1(K, Aφ[p]). Then there exist infinitely many admissible primes such that ∂`(s) = 0 and v`(s)6= 0.
Proof. — A direct generalization of [4, Theorem 3.2].
2.2. Controlling the Selmer group
Definition 2.4. — A collection of cohomology classes {κ`}`⊆H1(K`, Tφ[p])
indexed by the set of admissible primes`is said to be anEuler systemfor φ/K relative topif each classκ`enjoys the following properties:
1. Forq-np`,∂q(κ`) = 0.
2. Forq|n−p,resq(κ`)∈Hord1 (Kq, Tφ[p]).
Note that in the previous definition no condition is required for the primes q | n+`. The next standard argument reduces the proof of The- orem C in the Introduction to the problem of producing an Euler system.
Lemma 2.5. — Let {κ`}` be an Euler system for φ/K relative to p and assume that for all but a finite number of primes ∂`(κ`) 6= 0. Then Selp(φ/K) = 0.
Proof. — Assume that there exists s ∈ Selp(φ/K) with s6= 0 and, by Theorem 2.3, fix an admissible prime`so thatres`(s)6= 0 and∂`(κ`)6= 0.
By the global reciprocity low of class field theory,P
qhresq(s),resq(κ`)iq = 0, where the sum is over all the prime ideals of OF. Proposition 1.8 and Definition 2.4 imply thatresq(s)andresq(κ`)are orthogonal to each other with respect to the local Tate pairingh , iq at primes q 6=`, so hres`(s), res`(κ`)i` = 0.Then since ∂`(κ`)6= 0, Lemma 2.2 yieldsres`(s) = 0. This
contradiction proves the lemma.
Theorem C of the Introduction now follows from the existence of an Euler system satisfying prescribed local conditions at`via Lemma 2.5. The rest of the work is devoted to the construction of an Euler system under the hypothesis that the order ats= 1of theL-functionLK(φ, s)ofφoverK is zero. The strategy will be to find the classes κ` for admissible primes
` in the cohomology of Shimura curves defined from quaternion algebras of discriminantn−`. The theory of congruences between modular forms at the admissible prime`will give the required properties. Before stating the result to be proved in the next sections, defineS to be the set of all the primesp >3 so that:
1. pdivides the algebraic partLK(φ)∈ZofLK(φ, s)defined in Sec- tion 5.1;
2. There are non-trivial congruences (modp) between φ and other eigenforms of leveln(for more details, see Definition 4.7);
3. The representationρφ,pis reducible.
The result to be proved in the next sections is the following:
Theorem 2.6. — Assume thatEis modular. Ifp6∈S, then there exists an Euler system{κ`}` forφrelative top.
3. The Jacquet-Langlands correspondence 3.1. Modular forms on quaternion algebras
Fix a quaternion algebraB overF of discriminantm−which is ramified at all the archimedean places ofF. Fix a prime idealm+prime tom− and letR⊆Bbe an Eichler order of levelm+. (Hence,m−=Qs
j=1qj is square- free andshas the same parity as [F :Q].) Denote byBb (respectively,R)b the finite adele ring ofB (respectively,R). Fix an open compact subgroup U ⊆Bb× and a ring C.
Definition 3.1. — The space ofC-valuedmodular formswith respect toU on the quaternion algebraB is theC-module
S2B(U;C) :=L(U;C)/L(U;C)triv
where the elements of L(U;C) are functions f : U\Bb×/B× → C and L(U;C)triv is the C-submodule of L(U;C) consisting of those functions which factor through the adelizationbnB : Bb →Fb of the norm mapnB : B→F.
For a fixed Eichler order R, refer toS2B(Rb×;C)as the space of modular formsof levelm+ and denote it byS2B(m+;C).
3.2. The Hecke algebraTBm+
The Hecke operatorTq
Letqbe a prime ideal ofOF not dividingm+and assume thatUq' R×q. Define the Hecke operator
Tq:Z[U\Bb×/B×]−→Z[U\Bb×/B×] by the rule: Tq(g) := P
α
1 α 0 πq
g+
πq 0 0 1
g, where the sum is extended over a set of representatives {α} for the quotient OF,q/q, the element πq ∈ OF,q is chosen so that valq(πq) = 1 and the matrices are ideles whoseq-component is the displayed one and the other compo- nents are all equal to 1. Now use strong approximation to describe Tq. Set:ObFh
1 q
i
:=ObF(q)Fq, andUh
1 q
i
:=U(q)GL2(Fq),where the superscript
(q) denotes the idele with the q-component removed. By the strong ap- proximation theorem (see [47, page 60]: Bb× ' `h
j=1Uh
1 q
i
gjB×, where g1, . . . , gh are representatives of the double coset spaceU(q)\Bb(q)×/B× = Uh
1 q
i\Bb×/B× ' bnB Uh
1 q
i\Fb×/F+ (the last isomorphism is induced by the adelizationbnB of the norm map andF+ denotes the group of to- tally positive elements ofF). Define now the following subgroups for each j= 1, . . . , h:
eΓj,0,q:=g−1j U 1
q
gj∩B×; Γj,0,q:=Γej,0,q/OF
1 q
×
;
˜Γj,+,q:= (˜Γj,0,q)e; Γj,+,q:= (Γj,0,q)e,
where the subscriptemeans elements of evenq-adic valuation. The strong approximation theorem yields:
U\Bb×/B×'
h
a
j=1
PGL2(OF,q)\PGL2(Fq)(gj)q/Γj,0,q
where (gj)q denotes the projection of gj on the q-component. Note that g∈U\Bb×/B×lies on thei-th component of the last product if and only if bnB(g) =bnB(gi), soTq acts componentwise. Denote byTq the Bruhat-Tits tree ofPGL2(Fq)and by V(Tq)its vertexes. For eachj there is a natural projection V(Tq) → PGL2(OF,q)\PGL2(Fq)(gj)q/Γj,0,q. The operator Tq is induced by the projection of the operatorTeqofZ[V(Tq)]which associates to each vertexvthe sum of the verticeswwhose distance fromvis 1. Note that the degree ofTq is|q|+ 1.
The Hecke operatorUq
Let q be a prime ideal of OF dividing m+ and assume that Uq ' R×q. Using the same conventions as above, define the Hecke operator
Uq:Z[U\Bb×/B×]−→Z[U\Bb×/B×] by the rule:Uq(g) :=P
α
1 α 0 πq
g,where{α}is a set of representatives forOF,q/q. Suppose that q divides m+ exactly. In this case it is possible to obtain a description for Uq as for Tq. Using the same notations and definitions as above, the strong approximation theorem yields:
U\Bb×/B× =
h
a
j=1
Γ0(πq)\PGL2(Fq)/Γj,0,q,
whereΓ0(πq)⊆PGL2(OF,q)is the subgroup of (modq)upper triangular matrices. For eachjthere is a projection map−→
E(Tq) = Γ0(πq)\PGL2(Fq)→ Γ0(πq)\PGL2(Fq)/Γj,0,q.The operator Uq is induced by the projection of the operatorUeqofZ[E(Tq)]which associates to any oriented edgee= (v, w) the sum of the edgese0emanating from its target (that is, of the form(w, z) for some vertexz6=v). It is clear that the degree ofUq is|q|.
Hecke algebras
For a prime ideal q-m+m− so thatUq'R×q, denote by Sq:Z[U\Bb×/B×]→Z[U\Bb×/B×] the spherical operator defined by:Sq(g) :=P
ααg,whereU πqU =P
ααU.
Denote byTBm+ the freeZ-algebra generated byTqandSq for prime ideals q - m+m− and Uq for prime ideals q | m+. The space of modular forms S2B(m+;Z) is naturally a TBm+-module. Moreover, if Uq ' Rq for q 6∈ S, whereS is a finite set of prime ideals, thenS2B(U;Z)is naturally aTB,(S)m+ - module, whereTB,(S)m+ is the subalgebra of TBm+ generated by Hecke opera- tors corresponding to primes not inS.
Generalized Brandt matrices
It is possible to give another description of Hecke operators in terms of Brandt matrices. This is a classical subject and an example of the Jacquet- Langlands correspondence, established in this case by the Eichler trace
