Counting local systems with principal unipotent local monodromy

Let X₁ be a curve of genus g, projective and smooth over F_q. 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 F_q. 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 Σn_iγ^m_i for suitable integers n_i and "eigenvalues" γ_i. 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.