Mathematical Research Letters

Volume 24 (2017)

Number 6

On the number of irreducible mod $\ell$ rank $2$ sheaves on curves over finite fields

Pages: 1605 – 1632

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n6.a2

Authors

Gebhard Böckle (Interdisciplinary Center for Scientific Computing, Universität Heidelberg, Germany)

Chandrashekhar B. Khare (Department of Mathematics, University of California at Los Angeles)

Abstract

Let $X$ be a smooth, geometrically connected, projective curve of genus g over a finite field $\mathbb{F}_q$ of characteristic p. Consider primes $\ell$ different from $p$. We formulate some questions related to a well known counting formula of Drinfeld. Drinfeld counts rank $2$, irreducible $\ell$-adic sheaves on $X_n = X \times {}_{\mathbb{F}_q} \: \mathbb{F}_{q^n}$ as $n$ varies. We would like to count rank $2$, irreducible mod $\ell$ sheaves on $X_n$ as $n$ varies. Drinfeld’s $\ell$-adic count gives an upper bound for the mod $\ell$ count. We conjecture that Drinfeld’s count is the correct asymptotic for the count of rank $2$, irreducible mod $\ell$ sheaves on $X_n$ as $n$ varies, and $(n, \ell) = 1$. The conjecture is an invitation to finding good lower bounds on the number of irreducible rank $2$, mod $\ell$ sheaves on $X_n$.

We produce a lower bound on the mod $\ell$ count, which is weaker than the one conjectured, by counting “dihedral” mod $\ell$ sheaves. We make a deformation theoretic study of the number of $\ell$-adic sheaves which lift the pull backs $\mathcal{F}_n$ to $X_n$, of a given mod $\ell$ sheaf $\mathcal{F}_{n_0}$ on $X_{n_0}$, as we ranger over $n$ with $n_0 \vert n$.

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G.B. would like to thank the University of California at Los Angeles for hospitality where parts of this work was carried out. G.B. was supported with the DFG programs FG 1920 and SPP 1489 (in a joint project with FNR Luxembourg). C.K. would like to thank TIFR and Universität Heidelberg for their hospitality, and acknowledge support from a Humboldt Research Award and NSF grants, during the work on the paper.

Received 31 December 2016

Published 29 January 2018