Metaplectic cusp forms and the large sieve

Abstract

We prove a power saving upper bound for the sum of Fourier coefficients $\rho_f(\cdot)$ of a fixed cubic metaplectic cusp form $f$ over primes. Our result is the cubic analogue of a celebrated 1990 Theorem of Duke and Iwaniec, and the cuspidal analogue of a Theorem due to the author and Radziwill for the bias in cubic Gauss sums. The proof has two main inputs, both of independent interest. Firstly, we prove a new large sieve estimate for a bilinear form whose kernel function is $\rho_f(\cdot)$. The proof of the bilinear estimate uses a number field version of circle method due to Browning and Vishe, Voronoi summation, and Gauss-Ramanujan sums. Secondly, we use Voronoi summation and the cubic large sieve of Heath-Brown to prove an estimate for a linear form involving $\rho_f(\cdot)$. Our linear estimate overcomes a bottleneck occurring at level of distribution $2/3$.