Galois Symbols on the square of an elliptic curve

by

Jacob Murre and Dinakar Ramakrishnan^{∗}

Abstract

We prove some theorems concerning the Galois symbol map

s_{p} : T_{F} (E × E)/p → H^{2}(F, E[p]^{⊗2}),

where F is a local or a number field, E a semistable elliptic curve over F, p a prime, E[p] the Galois module of p-division points in E(\overline{F}), and T_{F} (E × E) the Albanese kernel over F. Explicitly, T_{F} (E × E) is generated by norms from finite extensions K/F of symbols ⟨P, Q⟩ representing the linear equivalence class of (P, Q) − (P, 0) − (0, Q) + (0, 0), with P, Q ∈ E(K).

When F is a number field, our global results include the following:

(i) The image of s_{p} is zero for any odd prime p which is unramified in F;

(ii) If p is an odd prime such that E has good ordinary reduction at a place of F above p (or if p = 2 with F having a real place), then for a suitable quadratic extension K of F ([E[p]) there exists an element θ in T_{K}(E × E)/p such that s_{p}(θ) is non-zero.

On the way we also prove (for p odd) that if E has ordinary or multiplicative reduction, then s_{p} is injective, and we describe the image precisely. When E has supersingular reduction, we get sufficient conditions for the image to be zero. Over archimedean fields, s_{p} is injective for all p, and the image is simple to describe.

In order to tie together the local results and get the global one, we establish the following local-global result:

Let Σ denote the set of all places v of F. Then the natural map

H^{2}(F, E[p]^{⊗2}) → ∏_{v∈Σ} H^{2}(F_{v}, E[p]^{⊗2})

is injective.

Jacob Murre  Dinakar Ramakrishnan
Department of Mathematics Department of Mathematics
University of Leiden  California Institute of Technology
Leiden, Netherlands  Pasadena, CA 91125, USA

∗Partially supported by the NSF