Zeta-functions of A-fields

Abstract

From now on, k will be an A-field of any characteristic, either 0 or p>1. Notations will be as before; if v is a place of k, k _v is the completion of k at v; if v is a finite place, r _v is the maximal compact subring of k _v , and p _v the maximal ideal in r _v . Moreover, in the latter case, we will agree once for all to denote by q _v the module of the field k _v and by π _v a prime element of k _v , so that, by th. 6 of Chap. I–4, r _v /p _v is a field with q _v elements, and |π _v | _v = q ^-1 _v . If k is of characteristic p>1, we will denote by q the number of elements of the field of constants of k and identify that field with F _q ; then, according to the definitions in Chap. VI, we have q _v = q ^deg(v) for every place v.