On the action of automorphisms of a curve on the first $l$-adic cohomology

Abstract

Introduction Let G= be a cyclic group with w = #G>l. For each divisor d of n, let %d denote the character of G of the irreducible representation over Q, the field of rational numbers, whose kernel is equal to (,od). Let k be an algebraicallyclosed ground field and X be a complete non-singular curve over k of genus Q^Z.2.The main purpose of the present paper is to prove the following theorems, under the assumption that G^=Aut(X), the automorphism group of X; in this situation we denote by Tx(G\H1(X, QO) tne character of the natural representation of G on the first/-adic cohomolqgy .fiPCX,Qz) of X, where I is any prime number differentfrom the characteristicof k fcf. Notation, 5 2 and also [41) :