2013/05/06 by Herman, P. Edward
#11D25 #11F70 #11F72 #11L05 #11L07 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1305.1243
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 Livné and Patterson \citeLP on sums of cubic exponential sums over \Q[ω], ω3=1. The key ideas in getting such an asymptotic are a Kuznetsov-like trace formula for metaplectic forms over a quartic cover of GL2, 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 \citeP.