On p-rank representations

Abstract

The p-rank of an algebraic curve X over an algebraically closed field k of characteristic p>0 is the dimension of the first etale cohomology vector space H^1(X,Z/pZ). We study the representations of finite groups G of automorphisms of X induced on the base extension of this vector space to k, and obtain two main results: First, the sum of the nonprojective direct summands of the representation, i.e. its core, is determined explicitly by local data given by the fixed point structure of the group acting on the curve. As a corollary, we derive a congruence formula for the p-rank. Secondly, the multiplicities of the projective direct summands of quotient curves, i.e. their Borne invariants, are calculated in terms of the Borne invariants of the original curve and ramification data. In particular, this is a generalization of both Nakajima's equivariant Deuring-Shafarevich formula and a previous result of Borne in the case of free actions.