Lattice points on determinant surfaces and the spectrum of the automorphic Laplacian

TL;DR

This paper derives an asymptotic count with error estimates for integer points on a determinant surface within expanding regions, employing advanced analytical techniques from number theory and harmonic analysis.

Contribution

It introduces a novel asymptotic formula for lattice points on determinant surfaces using a combination of Poisson summation, stationary phase, and spectral methods.

Findings

Established an explicit asymptotic count for lattice points on the surface

Provided error bounds for the counting formula

Applied advanced harmonic analysis techniques to number theory problems

Abstract

We obtain an asymptotic formula with an error term for counting integer points $(a, b, c, d)$ in an expanding box $ [-X, X]^4$ that lie on the determinant surface $xy-zw=r$ for $r\neq 0$. The method involves Poisson summation formula, stationary phase analysis, Kuznetsov formula, Weyl law and large sieve inequality for the twisted coefficients $\rho_j(n)n^{i\kappa_j}$.