Euler characteristics forp-adic Lie groups

Abstract

Lazard [23] found definitive results about the cohomology of p-adic Lie groups such as GLnZp with coefficients in vector spaces over Qp. These results, applied to the image of a Galois representation, have been used many times in number theory. It remains a challenge to understand the cohomology of p-adic Lie groups with integral coefficients, and especially to relate the integral cohomology of these groups to the cohomology of suitable Lie algebras over the p-adic integers Zp. In this paper, we do enough in this direction to compute a subtle version of the Euler characteristic, arising in the number-theoretic work of Coates and Howson ([14], [13]), for most of the interesting p-adic Lie groups. The Euler characteristics considered in this paper have the following form. Let G be a compact p-adic Lie group with no p-torsion. Let M be a finitely generated Zp-module on which G acts, and suppose that the homology groups Hi(G,M) are finite for all i. They are automatically 0 for i sufficiently large [27]. Then we want to compute the alternating sum of the p-adic orders of the groups Hi(G,M):