Local Galois Symbols on E × E
نویسندگان
چکیده
This article studies the Albanese kernel TF (E × E), for an elliptic curve E over a p-adic field F . The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from TF (E ×E)/p to the Galois cohomology group H(F, E[p] ⊗ E[p]), for E/F ordinary, without requiring that the ptorsion points are F -rational, or even that the Galois module E[p] is semisimple. A key step is to show that the image is zero when the finite Galois module E[p] is acted on non-trivially by the pro-p-inertia group Ip. Non-trivial classes in the image are also constructed when E[p] is suitably unramified. A forthcoming sequel will deal with global questions.
منابع مشابه
Local Galois Symbols on E ×
This article studies the Albanese kernel TF (E × E), for an elliptic curve E over a p-adic field F . The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from TF (E ×E)/p to the Galois cohomology group H(F, E[p] ⊗ E[p]), for E/F ordinary, without requiring that the ptorsion points are F -rational, or even that the Galois module E[p] is ...
متن کاملOn the Dimension of the Space of Cusp Forms Associated to 2-dimensional Complex Galois Representations
The aim of this paper is to use the “amplification technique” to obtain estimates on the dimension of spaces of automorphic forms associated to Galois representations; these bounds improve nontrivially on the work of Duke ([D]). A cuspidal representation π of GL2(AQ) is associated to a 2-dimensional Galois representation ρ : Gal(Q/Q) → GL2(C) if, for each place v, the local representation πv is...
متن کاملGalois Extensions of Hilbertian Fields
We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all ∈ G(K)e, the field Ks[ ] ∩Ktot,S is PSC. Here a local prime is an equivalent class p of absolute values of K whose completion is a local field, K̂p. Then Kp = Ks ∩ K̂p and Ktot,S = T p∈S T σ∈G(K) K σ p . G(K) stands for the absolute Galois ...
متن کاملDeformation of Outer Representations of Galois Group II
This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained several universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for...
متن کاملA refined conjecture of Mazur-Tate type for Heegner points
In [MT1], B. Mazur and J. Tate present a “refined conjecture of Birch and Swinnerton-Dyer type” for a modular elliptic curve E. This conjecture relates congruences for certain integral homology cycles on E(C) (the modular symbols) to the arithmetic of E over Q. In this paper we formulate an analogous conjecture for E over suitable imaginary quadratic fields, in which the role of the modular sym...
متن کامل