Variance of sums in arithmetic progressions of divisor functions associated with higher degree l-functions in $\mathbb{F}_q(t)$

TL;DR

This paper calculates the variances of sums of generalized divisor functions linked to higher degree L-functions over function fields, revealing new behaviors when the degree exceeds one, with applications to elliptic curves.

Contribution

It extends previous results on divisor functions by analyzing higher degree L-functions in function fields, using equidistribution and matrix integrals to compute variances.

Findings

Variance expressed as matrix integrals for large q

New qualitative behaviors for L-functions of degree > 1

Application to elliptic curves over function fields

Abstract

We compute the variances of sums in arithmetic progressions of generalised k-divisor functions related to certain L-functions in $\mathbb{F}_q(t)$, in the limit as $q\to\infty$. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when $q\to\infty$, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual $k$-divisor function, when the L-function in question has degree one. They illustrate the role played by the degree of the L-functions; in particular, we find qualitatively new behaviour when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over $\mathbb{F}_q(t)$, and we illustrate them by examining in some detail the generalised $k$-divisor functions associated with the Legendre curve.