Counting local systems with principal unipotent local monodromy

Abstract

Let X1 be a curve of genus g, projective and smooth over Fq. Let S1 sub; 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 prime l not dividing q. The pullback by the Frobenius endomorphism Fr of X 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 s ε S is unipotent, with a single Jordan block. Let T(X1; S1; n,m) be the number of fixed 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 fixed point formula: the function m→T(X1; S1; n,m) is Σniγmi for suitable integers ni and "eigenvalues" γi. We use Lafforgue to reduce the computation of T(X1; S1; n,m) to countingv automorphic representations of GL(n), and the assumption N1≥2 to move the counting to the multiplicative group of a division algebra, where the trace formula is easier to use. © 2013 Department of Mathematics, Princeton University.