On Patterson's Conjecture: Sums of Quartic Exponential Sums

TL;DR

This paper provides new asymptotic results for quartic exponential sums over Gaussian integers, offering evidence for Patterson's conjecture and revealing unexpected secondary main terms.

Contribution

It introduces a Kuznetsov-like trace formula for quartic covers of GL(2) and an exponential sum identity, advancing understanding of Patterson's conjecture over number fields.

Findings

Asymptotic formula for sums of quartic exponential sums over .

Identification of a secondary main term in the asymptotic expansion.

Extension of spectral techniques to quartic exponential sums.

Abstract

We give more evidence for Patterson's conjecture on sums of exponential sums, by getting an asymptotic for a sum of quartic exponential sums over $\Q[i].$ Previously, the strongest evidence of Patterson's conjecture over a number field is the paper of Livn\'{e} and Patterson \cite{LP} on sums of cubic exponential sums over $\Q[\omega], \omega^3=1.$ The key ideas in getting such an asymptotic are a Kuznetsov-like trace formula for metaplectic forms over a quartic cover of $GL_2,$ and an identity on exponential sums relating Kloosterman sums and quartic exponential sums. To synthesize the spectral theory and the exponential sum identity, there is need for a good amount of analytic number theory. An unexpected aspect of the asymptotic of the sums of exponential sums is that there can be a secondary main term additional to the main term which is not predicted in Patterson's original paper \cite{P}.