Vanishing of some Galois cohomology groups for elliptic curves
نویسندگان
چکیده
Let E/Q be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of Q obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group H ( G,E[p] ) does not vanish, and investigate the analogous question for E[p] when i > 1. We include an application to the verification of certain cases of the Birch and SwinnertonDyer conjecture, and another application to the Grunwald–Wang problem for elliptic curves.
منابع مشابه
Vanishing of Some Cohomology Groups and Bounds for the Shafarevich-Tate Groups of Elliptic Curves
Let E be an elliptic curve over Q and ` be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at `. Under certain assumptions, we prove the vanishing of the Galois cohomology group H1(Gal(K(E[`i])/K), E[`i]) for all i ≥ 1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyv...
متن کاملSelmer groups and Mordell-Weil groups of elliptic curves over towers of function fields
In [12] and [13], Silverman discusses the problem of bounding the Mordell-Weil ranks of elliptic curves over towers of function fields. We first prove generalizations of the theorems of those two papers by a different method, allowing non-abelian Galois groups and removing the dependence on Tate’s conjectures. We then prove some theorems about the growth of Mordell-Weil ranks in towers of funct...
متن کاملGalois Groups of Difference Equations of Order Two on Elliptic Curves
This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations wi...
متن کاملDiagonal Cycles and Euler Systems Ii: the Birch and Swinnerton-dyer Conjecture for Hasse-weil-artin L-functions
This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over Q viewed over the fields cut out by certain self-dual Artin representations of dimension at most 4. When the associated L-function vanishes (to even order ≥ 2) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be lin...
متن کاملHILBERT MODULAR FORMS AND p-ADIC HODGE THEORY
We consider the p-adic Galois representation associated to a Hilbert modular form. Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a place not dividing p is compatible with the local Langlands correspondence [C2]. In this paper, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory [Fo], as is shown for an e...
متن کامل