Invent. math. 124, 129–174 (1996) Kummer theory for abelian varieties over local elds J. Coates 1 , R. Greenberg 2;? 1 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK 2 Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA Oblatum 29-III-1995 & 12-IV-1995 to Reinhold Remmert 1 Introduction This paper attempts to throw new light on two classical questions about the arithmetic of abelian varieties over local elds. Let p be any prime number, Q p the eld of p-adic numbers, and Q p a xed algebraic closure of Q p . Let A be an abelian variety, which is dened over a nite extension F of Q p lying inside of Q p . As usual, we let A[p ∞ ] denote the p-primary subgroup of A( Q p ). It is endowed with a natural action of the Galois group G F = G( Q p =F ). Now let K be any extension of F contained in Q p . We recall that the classical Kummer homomorphism (1:1)  A;K : A(K ) ⊗ Q p = Z p → H 1 (K;A[p ∞ ]) is dened by mapping P ⊗ (p −n mod Z p ) (where P ∈ A(K ), n = 0) to the class of the 1-cocycle ’ dened by ’()= (Q) − Q for all  ∈ G K ; here Q is any point in A( Q p ) such that p n Q = P. The rst and main problem we shall be concerned with in this paper is: (I) To nd a description of the image of  A;K solely in terms of the G F -module A[p ∞ ]. The main earlier work on (I) is due to Bloch and Kato ([1], Sect. 3), who provided an answer to it for all nite extensions K of F , by using Fontaine’s mysterious ring B DR . This is a highly important result. In this article, we shall be primarily concerned with innite extensions of F . Our approach has been partly motivated by the earlier celebrated paper of Tate [18]. By generalizing the arguments of Sect. 3 of [18], we have been led to introduce a new class of innite algebraic extensions K of Q p , which we call deeply ramied. The ? Supported partially by a National Science Foundation grant