Counting rank two local systems with at most one, unipotent, monodromy

Abstract

The number of rank two $\overline{\Bbb{Q}}_\ell$-local systems, or $\overline{\Bbb{Q}}_\ell$-smooth sheaves, on $(X-\{u\})\otimes_{\Bbb{F}_q}\Bbb{F}$, where $X$ is a smooth projective absolutely irreducible curve over $\Bbb{F}_q$, $\Bbb{F}$ an algebraic closure of $\Bbb{F}_q$ and $u$ is a closed point of $X$, with principal unipotent monodromy at $u$, and fixed by ${\rm Gal}(\Bbb{F}/\Bbb{F}_q)$, is computed. It is expressed as the trace of the Frobenius on the virtual $\overline{\Bbb{Q}}_\ell$-smooth sheaf found in the author's work with Deligne on the moduli stack of curves with \'etale divisors of degree $M\ge 1$. This completes the work with Deligne in rank two. This number is the same as that of representations of the fundamental group $\pi_1((X-\{u\})\otimes_{\Bbb{F}_q}\Bbb{F})$ invariant under the Frobenius ${\rm Fr}_q$ with principal unipotent monodromy at $u$, or cuspidal representations of ${\rm GL}(2)$ over the function field $F=\Bbb{F}_q(X)$ of $X$ over $\Bbb{F}_q$ with Steinberg component twisted by an unramified character at $u$ and unramified elsewhere, trivial at the fixed id\`ele $\alpha$ of degree 1. This number is computed in Theorem 4.1 using the trace formula evaluated at $f_u\prod_{v\not=u}\chi_{K_v}$, with an Iwahori component $f_u=\chi_{I_u}/|I_u|$, hence also the pseudo-coefficient $\chi_{I_u}/|I_u|-2\chi_{K_u}$ of the Steinberg representation twisted by any unramified character, at $u$. Theorem 2.1 records the trace formula for ${\rm GL}(2)$ over the function field $F$. The proof of the trace formula of Theorem 2.1 recently appeared elsewhere. Theorem 3.1 computes, following Drinfeld, the number of $\overline{\Bbb{Q}}_\ell$-local systems, or $\overline{\Bbb{Q}}_\ell$-smooth sheaves, on $X\otimes_{\Bbb{F}_q}\Bbb{F}$, fixed by ${\rm Fr}_q$, namely $\overline{\Bbb{Q}}_\ell$-representations of the absolute fundamental group $\pi(X\otimes_{\Bbb{F}_q}\Bbb{F})$ invariant under the Frobenius, by counting the nowhere ramified cuspidal representations of ${\rm GL}(2)$ trivial at a fixed id\`ele $\alpha$ of degree 1. This number is expressed as the trace of the Frobenius of a virtual $\overline{\Bbb{Q}}_\ell$-smooth sheaf on a moduli stack. This number is obtained on evaluating the trace formula at the characteristic function $\prod_v\chi_{K_v}$ of the maximal compact subgroup, with volume normalized by $|K_v|=1$. Section 5, based on a letter of P. Deligne to the author dated August 8, 2012, computes the number of such objects with any unipotent monodromy, principal or trivial, in our rank two case. Surprisingly, this number depends only on $X$ and ${\rm deg}(S)$, and not on the degrees of the points in $S_1$.