Metaplectic cusp forms and the large sieve
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917067911856128 |
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| author | Dunn, Alexander |
| author_facet | Dunn, Alexander |
| contents | 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 $2/3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13151 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metaplectic cusp forms and the large sieve Dunn, Alexander Number Theory 11F27, 11F30, 11L05, 11L20, 11N36 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 $2/3$. |
| title | Metaplectic cusp forms and the large sieve |
| topic | Number Theory 11F27, 11F30, 11L05, 11L20, 11N36 |
| url | https://arxiv.org/abs/2403.13151 |