On Patterson's Conjecture: Sums of Quartic Exponential Sums

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 $\text{Q}[i]\mathrm{.}$ 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 $\text{Q}[\omega ],{\omega }^3=1.$ The key ideas in getting such an asymptotic are a Kuznetsov-like trace formula for metaplectic forms over a quartic cover of $G{L}_2,$ 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}.