Counting local systems with principal unipotent local monodromy

Abstract

Let X1 be a curve of genus g, projective and smooth over Fq. Let S1 X1 be a reduced divisor consisting of N1 closed points of X1. Let (X;S) be obtained from (X1;S1) by extension of scalars to an algebraic closure F of Fq. Fix a primel not dividingq. The pullback by the Frobenius endomorphism Fr ofX induces a permutation Fr of the set of isomorphism classes of rank n irreducible Ql-local systems on X S. It maps to itself the subset of those classes for which the local monodromy at each s2 S is unipotent, with a single Jordan block. Let T (X1;S1;n;m) be the number of xed points of Fr m acting on this subset. Under the assumption that N1 2, we show that T (X1;S1;n;m) is given by a formula reminiscent of a Lefschetz xed point formula: the function m 7! T (X1;S1;n;m) is of the form P ni m for suitable integers ni and \eigenvalues i. We use Laorgue