Iwasawa Theory for Deformation of Ordinary Elliptic Curves, Study notes of Mathematics

Download Iwasawa Theory for Deformation of Ordinary Elliptic Curves and more Study notes Mathematics in PDF only on Docsity! Iwasawa theory for deformation of ordinary elliptic curves 1 Special values of L-functions of ordinary elliptic curves 1.1 Review of elliptic curves Let E be an elliptic curve over Q of conductor N . Let K be an imaginary quadratic extension of Q of discriminant dK lesser than 4 and prime to N and let ηK be the quadratic character of Gal(Q̄/Q) attached to K. Let p ≥ 5 be a rational prime which does not divide NdK 1.1.1 Geometric data Let GK be Gal(K̄/K). The group E(K̄) being divisible, there is an exact sequence 0 −→ E(K̄)[pn] −→ E(K̄) p n −→ E(K̄) −→ 0 and thus an exact sequence in GK-cohomology: 0 −→ E(K)/pnE(K) −→ H1(GK , E[pn]) −→ H1(GK , E)[pn] −→ 0 Doing the same with all the completions of K at finite places, we obtain a commutative diagram: 0 // E(K)/pnE(K) //  H1(GK , E[pn]) //  H1(GK , E)[pn] //  0 0 // ∏ v E(Kv)/pnE(Kv) // ∏ v H1(GKv , E[p n]) // ∏ v H1(GKv , E)[p n] // 0 Let Seln(E/K) and X(E/K) be: Seln(E/K) = ker ( H1(GK , E[pn]) −→ ∏ v H1(GKv , E) ) X(E/K) = ker ( H1(GK , E) −→ ∏ v H1(GKv , E) ) Using the snake lemma, we see that there is a short exact sequence 0 −→ E(K)⊗ Z/pnZ −→ Seln(E/K) −→X(E/K)[pn] −→ 0 which is also interesting after taking direct and inverse limit on n: 0 −→ E(K)⊗Qp/Zp −→ Sel∞(E/K) −→X(E/K)[p∞] −→ 0 0 −→ E(K)⊗ Zp −→ Sel∞(E/K) −→ TpX(E/K) −→ 0 1 1.1.2 Automorphic data The reduction modulo a prime ` of E/Q defines a curve E/F`, which is elliptic for primes ` not dividing N . Let a` be 1 + `−#E/F` and P`(X) be 1− a`X + `X2 if ` - N and or 1− a`X if `|N . The local L-function L` is then defined to be: L`(E/Q, s) = 1 P`(`−s) The product of local L-functions defines the complex L-function: L(E/Q, s) = ∏ ` L`(E/Q, s) = ∏ `-N 1 1− a``−s + `1−2s ∏ `|N 1 1− a``−s By [Wil95, BCDT01], the complex L-function L(E/Q, s) coincides with the Mellin transform of a modular form f ∈ S2(Γ0(N)) which is an eigenvector under the action of the Hecke operators T` for ` - N . The defining property of invariance under Γ0(N) of modular forms implies that their Mellin transforms satisfy functional equations. Hence, L(E/Q, s) satisfies the functional equation: L(E/Q, s) = εC(s)L(E/Q, 2− s) The factor C(s) comes from the local factors at∞ of L(E/Q, s) and L(E/Q, 2−s). Because C(1) is equal to 1 and because we are only interested in the order of vanishing of L(E/Q, 1) at 1, we ignore it in the following. The ε-factor ε is equal to ±1 and can be defined and computed locally, though computations are actually quite involved when π(f)` is supercuspidal. The important property for what follows is that ε` is equal to 1 for ` - N∞ and that ε` only depends on E/Q`. We will also consider the twisted L-function L(E, ηK , s) defined by: L(E, ηK , s) = ∏ `-N 1 1− a`ηK(`)`−s + `1−2s ∏ `|N 1 1− a`ηK(`)`−s The L-function of the elliptic curve E/K is defined to be: L(E/K, s) = L(E/Q, s)L(E/Q, ηK , s) This L-function is the automorphic L-function of the base change πK(f) of π(f) to AK as well as the Rankin-Selberg L-function π(f)× θ(χtriv). It satisfies the functional equation: L(E/K, s) = εKCK(s)L(E/K, 2− s) We ignore the contribution of CK(s) for the same reasons as above. The ε factor εK is equal to: εK = (−1)#S , S = {`|εK,` 6= ηK,`(−1)} The values of the local εK,` factors is summed up below: 1. Assume πK(f)` is a principal series. Then εK,` = ηK,`(−1) (hence ` /∈ S). 2. Assume πK(f)` = St(µ) is Steinberg. Then εK,` = −ηK,`(−1) if ` is not split in K and µ = 1, else it is equal to ηv(−1) (hence ` belongs to S if and only if ` is inert in K and ord`N is odd). 3. Assume πK(f)` is supercuspidal. Then εK,` = η`(`)ord` N (Hence ` belongs to S if and only if ` is inert in K and ord`N is odd). 4. The ε-factor at ∞ is equal to 1 (hence ∞ belongs to S). We assume henceforth that all primes dividing N split in K. Then, εK is equal to −1 so the L-function L(E/K, s) vanishes at odd, and hence non-zero, order at 1. 2 And: THi = HomZp(e ord m J∞, µp∞)⊗Tordm R (= lim←− s eordm H 1 et(X(N, s)×Q Q̄,Zp)⊗Tordm R) The R-module THi is free of rank 2. We assume that R is a Gorenstein ring (this means for example that HomΛ(R,Λ) is a free R-module of rank 1). In the talk, I unfortunately claimed that R is a Gorenstein, but as Professor Hida remarked, this is not known to be true. Arithmetic specializations of THi are the GQ-representations attached to ordinary eigenforms with same residual representation as E in the sense that for ` - Np, the GQ-representation THi verifies det(1− Fr(`)X) = 1− λ(T`)X + χΓ(`)`X2 where λ is the natural morphism from Tord to R and χΓ is the inclusion of Γ inside R. 2.3 Combining the two Let RIw be R[[Γa]] and T be THi ⊗R RIw with GK-action on both side of the tensor product. The module T is free of rank 2 over a Gorenstein ring of Krull dimension 3. In fact, we will consider self-dual twists of THi and T . The determinant of THi verifies: detRTHi = R(1)⊗ χΓ The character χΓ is a square because 1+pZp is 2-divisible. Choose χ such that χ2 = χΓ and define: T †Hi = THi ⊗ χ −1, T † = T ⊗ χ−1 Then THi and T † are self-dual. Note that an arithmetic specialization of THi of even weight k is equal to the (cohomological) GQ-representation of f twisted by k/2. 3 A question For T equal to TIw or to T † Hi or to T †, is it possible to formulate and prove equivalents of (1.2.1) and of (1.2.2) for the GK-representation T . Stating a generalization of (1.2.1) would entail the following: 1. Define a generalization H̃1f (K,T ) of the group of rational points and H̃ 2 f (K,T ) of X for T . Presumably, these should be subgroups of Galois cohomology with coefficients in the ring of coefficients of T and we would expect H̃1f (K,T ) to be of rank 1. 2. Define a generalization z of Heegner points in this setting. Presumably, z should belong to H̃1f (K,T ), this is to say to the generalization of the group of rational points. 3. The equation (1.2.2) involves the cardinal of X and of E(K)/z. It doesn’t seem reasonable to hope for our more general H̃if (K,T ) to be of finite cardinality. Find a suitable generalization. In the setting described, everything in the above has in fact been achieved in [Gre91, Nek06, How07] when the ring of coefficients R is assumed to be regular, so automatically for TIw but under the supplementary hypothesis that R is a regular ring for T †Hi and T †. For general R, one can look up [Fou09]. As (1.2.1) stems for the Birch and Swinnerton-Dyer conjecture, it is moreover not unreasonable to make the vague requirement that H̃if (GK , T ) be linked with the special values of a suitable p-adic L-function. 5 4 Some progress towards the question 4.1 Selmer structures 4.1.1 Some motivation Since [BK90], it is known that a good equivalent of the group of K-rational points of an elliptic curve or an abelian variety for a general p-adic GK-representation can be constructed by looking at subgroups of Galois cohomology H1f (K,T ) ⊂ H1(K,T ) satisfying local conditions, in the sense that c ∈ H1(K,T ) belongs to H1f (K,T ) if and only if for all v, the local class cv ∈ H1(Kv, T ) verifies some supplementary condition. Among the conditions we can impose on local cohomology, the following ones have proven useful: 1. The relaxed condition: cv verifies no supplementary condition. 2. The strict condition: cv is trivial. 3. The unramified condition: cv is trivial in H1(Iv, T ). 4. The ordinary or Greenberg condition at v|p: cv is trivial in H1(Kv, T−v ) for a given quotient T−v of T . 5. The crystalline or Bloch-Kato condition at v|p: cv is trivial in H1(Kv, V ⊗ Bcris) where V = T ⊗Qp. In general, the best equivalent of E(K) is given by the imposition of the unramified condition at v - p and of the Bloch-Kato condition at v|p (indeed, in that case, H1f (K,TpE) coincides with the image of E(K) inside H1(K,TpE)). However, the Bloch-Kato condition suffers from the defect that it is only defined at present for Qp-modules, so it cannot be used for our more general representations TIw or T †. 4.1.2 Selmer complexes Let T be TpE, TIw, T † Hi or T † Iw and let S be its ring of coefficients (hence always a Gorenstein domain). Let Σ be a finite set of places of K containing the places above Np and GK,Σ be the Galois group of the maximal extension of K unramified outside Σ. Let C•cont(GK,Σ, T ) be the complex of continuous cochains with values in T and for v ∈ Σ, let C•cont(GKv , T ) be the complex of local continuous cochains with values in T . If v|p, let Cf (GKv , T ) be the complex of continuous cochains C•cont(GKv , T+v ). If v belongs to Σ but v - p, let Cf (GKv , T ) be the complex of continuous cochains C•cont(GKv/Iv, T ). Remark that for all v ∈ Σ, there is a morphism of complexes iv from Cf (GKv , T ) to C•cont(GKv , T ) induced by the inclusion of T+v inside T if v|p and by inflation if v - p. Let R Γf (GK,Σ, T ) be the object in the derived category corresponding to: Cone ( C•cont(GK,Σ, T )⊕ ⊕ v∈Σ Cf (GKv , T ) resv −iv−→ ⊕ v∈Σ C•cont(GK,v, T ) ) [−1] We write H̃if (K,T ) for the i-th cohomology group of R Γf (GK,Σ, T ). This notation introduces no ambiguity because R Γf (GK,Σ, T ) is independent of the choice of Σ as long as it verifies what we required. Our generalization of E(K)⊗ Zp and X(E/K)[p∞] is H̃1f (K,T ) and H̃2f (K,T )tors. 6 4.2 The Iwasawa-theoretic case In this subsection, T is taken to be TIw. The ring of coefficients Λa being a 2-dimensional regular ring, for every Λa-moduleM , there is a pseudo-isomorphism (this is to say, in that case, a morphism with finite kernel and co-kernel) M  ∼−→ Λra ⊕ ⊕ p Λa/pn where the p are prime ideal of height one. If M is a torsion Λa-module, let charΛa M be the ideal generated by the products of the pn. Under the inspiration of B.Mazur, B.Perrin-Riou proposed in [PR87] the following conjecture: 1. The Λa-module H̃1f (K,TIw) is free of rank 1. 2. There exists a non-trivial class zIw in H̃1f (K,TIw) coming from Heegner points in lim←− n E(Dn). 3. There is the following equality of characteristic ideals: charΛa H̃ 2 f (K,TIw)tors = ( charΛa H̃ 1 f (K,TIw)/Λa · zIw )2 This generalizes neatly (1.2.1). Note that in this conjecture, it is not assumed that L(E/K, s) vanishes at order 1. This reflects a principle called Mazur’s conjecture that the generic order of vanishing of the L-function, hence the generic rank of H̃1f (K,T ) should be minimal. Thanks to [Cor02, CV04], the first two numbers of this conjecture has known. Thanks to [How04], it is known (the result is stated under slightly stronger hypotheses but ours are enough to prove it) that: charΛa H̃ 2 f (K,TIw)tors| ( charΛa H̃ 1 f (K,TIw)/Λa · zIw )2 4.3 The general regular case In this subsection, T is taken to be T † and R is assumed to be a regular ring (hence the same is true of RIw). The conjecture of the previous subsection has been generalized by B.Howard in [How07] to: 1. The RIw-module H̃1f (K, T †) is torsion-free of rank 1. 2. There exists a non-trivial class z∞ in H̃1f (K, T †) coming from Heegner points in the tower of modular curves lim ←− n lim ←− s X1(Nps)(Dn). 3. There is the following equality of characteristic ideals: charRIw H̃ 2 f (K, T †)tors = ( charRIw H̃ 1 f (K, T †)/RIw · z∞ )2 Moreover, B.Howard proved the first two numbers. A proof of the fact that charRIw H̃ 2 f (K, T †)tors| ( charRIw H̃ 1 f (K, T †)/RIw · z∞ )2 can be found in [Fou08]. 7