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arXiv · 2403.13151

Metaplectic cusp forms and the large sieve

Alexander Dunn
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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 $2/3$.

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Alexander Dunn. 2024-03-19. Metaplectic cusp forms and the large sieve. https://doi.org/10.2140/ant.2025.19.1823

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