Zeta-functions of A-fields
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 v - 1. 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.
KeywordsMeromorphic Function Haar Measure Standard Function Open Subgroup Finite Limit
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