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 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 Q l -local systems on X -S.It maps to itself the subset of those classes for which the local monodromy at each s ∈ 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 of the form niγ m i for suitable integers ni and "eigenvalues" γi.We use Lafforgue to reduce the computation of T (X1, S1, n, m) to counting 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.

Locations

• CiteSeer X (The Pennsylvania State University) - View - PDF
• Annals of Mathematics - View - PDF