Counting local systems with principal unipotent local monodromy


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.


Pierre Deligne

Institute for Advanced Study, Princeton, NJ

Yuval Z. Flicker

Ariel University, Ariel, Israel and
The Ohio State University, Columbus, OH