Number of irreducible mod l rank 2 sheaves on curves over finite fields

TL;DR

This paper explores the asymptotic behavior of counting irreducible rank 2 mod l sheaves on curves over finite fields, extending Drinfeld's l-adic counting formula to the mod l setting.

Contribution

It formulates conjectures relating Drinfeld's l-adic count to the actual count of irreducible mod l sheaves, proposing a new asymptotic formula.

Findings

Drinfeld's count provides an upper bound for mod l sheaves.

Conjecture that Drinfeld's count accurately predicts the asymptotic number of mod l sheaves.

Extension of counting formulas from l-adic to mod l sheaves over finite fields.

Abstract

Let X be a smooth projective curve of genus g over a finite field F_q of characteristic p. Consider primes l different from p. We formulate some questions related to a well known counting formula of Drinfeld. Drinfeld counts rank 2, irreducible l-adic sheaves on the base change X_n of X to F_{q^n} as n varies. We would like to count rank 2, irreducible mod l sheaves on X_n as n varies. Drinfeld's l-adic count gives an upper bound for the mod l count. We conjecture that Drinfeld's count is the correct asymptotic for the count of rank 2, irreducible mod l sheaves on X_n as n varies with (n,\ell)=1.