Distribution of integer points on determinant surfaces and a $\text{mod-}p$ analogue
Satadal Ganguly, Rachita Guria
TL;DR
The paper establishes asymptotic formulas with explicit main terms and strong error bounds for counting integer solutions to the determinant equation $xy-zw=r$ with smooth weights, and its mod-$p$ analogue $xy-zw\equiv1\pmod p$. The authors deploy a direct Poisson-summation approach that converts the counting problem into sums of Kloosterman sums, and then leverage the Kuznetsov trace formula alongside analytic bounds for automorphic forms (holomorphic, Maass, and continuous spectrum) to achieve tight error terms involving the Ramanujan–Petersson exponent $\theta$ (currently $\theta\le 7/64$ via Kim–Sarnak). A detailed decomposition of spectral contributions yields a bound $E_V(X,r) \ll r^{\theta} X^{1+\varepsilon}$ for the error, and the mod-$p$ analysis shows a main term of order $X^4/p$ with a controllable secondary error depending on a slowly growing function $g(p)$. These results illuminate determinant-surface point counts, connect to moments of $L$-functions, and provide robust smoothed and modular counts that enhance our understanding of representation by indefinite quadratic forms in four variables.
Abstract
We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form $xy-zw=r$, where $r$ is a non-zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables $x, y, z, w$ as well as of $r$. We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence $xy-zw \equiv 1 (\text{mod }p)$, where $p$ is a large prime, with a strong bound on the error term.