Lattice Point Variance in Thin Elliptic Annuli over $\mathbb{F}_q [T]$

TL;DR

This paper investigates the variance of a representation counting function over function fields, revealing detailed asymptotic behavior in the microscopic regime and extending methods involving Hankel matrices.

Contribution

It provides the first asymptotic formula for the variance of lattice point counts in thin elliptic annuli over function fields, especially in the local regime where the interval length is fixed.

Findings

Asymptotic variance formula in the microscopic regime

Behavior analysis at the boundary between short and long intervals

Extension of Hankel matrix techniques to new functions

Abstract

For fixed coprime polynomials $U,V \in \mathbb{F}_q [T]$ with degrees of different parities, and a general polynomial $A \in \mathbb{F}_q [T]$, define the arithmetic function $S_{U,V} (A)$ to be the number of representations of $A$ of the form $UE^2 + VF^2$ with $E,F \in \mathbb{F}_q [T]$. We study the mean and variance of $S_{U,V}$ over short intervals in $\mathbb{F}_q [T]$, and this can be interpreted as the function field analogue of the mean and variance of lattice points in thin elliptic annuli, where the scaling factor of the ellipses is rational. Our main result is an asymptotic formula for the variance even when the length of the interval remains constant relative to the absolute value of the centre of the interval. In terms of lattice points, this means we obtain the variance in the so-called ``local'' or ``microscopic'' regime, where the area of the annulus remains constant relative to the inner radius. We also obtain asymptotic or exact formulas for almost all other lengths of the interval, and we see some interesting behaviour at the boundary between short and long intervals. Our approach is that of additive characters and Hankel matrices that we employed for the divisor function and a restricted sum-of-squares function in previous work, and we develop further results on Hankel matrices in this paper.