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

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.