Metaplectic cusp forms and the large sieve

Dunn, Alexander

doi:10.2140/ant.2025.19.1823

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Metaplectic cusp forms and the large sieve

Alexander Dunn

Vol. 19 (2025), No. 9, 1823–1880

DOI: 10.2140/ant.2025.19.1823

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 Radziwiłł 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 $\frac{2}{3}$ .

Dedicated to Chantal David on the occasion of her 60th birthday.

Keywords

cubic metaplectic forms, primes, large sieve, circle method

Mathematical Subject Classification

Primary: 11F27, 11F30, 11L05, 11L20, 11N36

Milestones

Received: 21 March 2024

Revised: 8 July 2024

Accepted: 3 September 2024

Published: 21 July 2025

Authors

Alexander Dunn
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

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Vol. 19, No. 9, 2025