Euler-Poincaré characteristics of abelian varieties

Abstract

Let F be a finite extension of Q , and let A be an abelian variety defined over F. Let p be a prime, and let A_p^∞ be the Galois module of points of A of order a power of p. Let G denote the Galois group of the extension F(A_p^∞)/F. Suppose that G contains no element of order p. Then the cohomology groups Hⁱ(G, A_p^∞) are finite for all i ≥ 0, and zero for i sufficiently large; let ×(G, A_p^∞) be the alternating product of their orders. The principal result of this Note is that we have x(G, A_p^∞) = 1.