Counting local systems with principal unipotent local monodromy

Abstract

Let $X_1$ be a curve of genus $g$, projective and smooth over $\mathbb{F}_q$. Let $S_1\subset X_1$ be a reduced divisor consisting of $N_1$ closed points of $X_1$. Let $(X,S)$ be obtained from $(X_1,S_1)$ by extension of scalars to an algebraic closure $\mathbb{F}$ of $\mathbb{F}_q$. Fix a prime $l$ not dividing $q$. The pullback by the Frobenius endomorphism $\mathrm{Fr}$ of $X$ induces a permutation $\mathrm{Fr}^*$ of the set of isomorphism classes of rank $n$ irreducible $\overline{\mathbb{Q}}_l$-local systems on $X-S$. It maps to itself the subset of those classes for which the local monodromy at each $s\in S$ is unipotent, with a single Jordan block. Let $T(X_1,S_1,n,m)$ be the number of fixed points of $\mathrm{Fr}^{*m}$ acting on this subset. Under the assumption that $N_1{\,{\scriptstyle \ge}\,} 2$, we show that $T(X_1,S_1,n,m)$ is given by a formula reminiscent of a Lefschetz fixed point formula: the function $m\mapsto T(X_1,S_1,n,m)$ is of the form $\sum n_i\gamma_i^m$ for suitable integers $n_i$ and “eigenvalues” $\gamma_i$. We use Lafforgue to reduce the computation of $T(X_1,S_1,n,m)$ to counting automorphic representations of $\mathrm{GL}(n)$, and the assumption $N_1{\,{\scriptstyle \ge}\,} 2$ to move the counting to the multiplicative group of a division algebra, where the trace formula is easier to use.