Metaplectic cusp forms and the large sieve
Alexander Dunn
TL;DR
The paper establishes a power-saving bound for sums of cubic metaplectic Fourier coefficients over primes, mirroring the Duke–Iwaniec paradigm in the metaplectic setting. It achieves this via two principal pathways: a new large sieve for bilinear forms with kernel $\rho_f$, and a linear bound obtained through Voronoi summation coupled with Heath–Brown’s cubic large sieve to overcome a level-distribution bottleneck at $X^{2/3}$. Central technical tools include the Browning–Vishe circle method over number fields, a level-aspect Voronoi theory for twists, and intricate type-I/type-II estimates that are carefully balanced. The results yield explicit bounds for $\mathscr P_f$ and $\widetilde{\mathscr P_f}$ with precise exponents, advancing our understanding of prime-distribution phenomena for cubic metaplectic Fourier coefficients and contributing to the broader circle of automorphic form sieve methods. This work has potential implications for zero-free regions and bias phenomena in related $L$-functions and exponential sums in the cubic metaplectic context.
Abstract
We prove a power saving upper bound for the sum of Fourier coefficients $ρ_f(\cdot)$ 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 $ρ_f(\cdot)$. 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 $ρ_f(\cdot)$. Our linear estimate overcomes a bottleneck occurring at level of distribution $2/3$.