Counting integer points on quadrics with arithmetic weights

TL;DR

This paper establishes an upper bound for counting integer solutions to a specific quadratic form in four variables, weighted by Fourier coefficients of a holomorphic Hecke form, advancing understanding of weighted lattice point problems on quadrics.

Contribution

It provides a new upper bound for weighted counts of integer points on diagonal quadrics using Fourier coefficients, combining techniques from quadratic forms and automorphic forms.

Findings

Derived an explicit upper bound for weighted integer solutions

Extended methods to incorporate Fourier coefficient weights

Improved understanding of lattice point distribution on quadrics

Abstract

Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer zeros of $F$ with $|\boldsymbol{x}| \leq X$, weighted by $\lambda(x_1)$.