Non-abelian amplification and bilinear forms with Kloosterman sums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912702255857664 |
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| author | Pascadi, Alexandru |
| author_facet | Pascadi, Alexandru |
| contents | We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli $c$, using Fourier analysis on $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ and an amplification argument with non-abelian characters. For sums of length $\sqrt{c}$, our method produces a non-trivial bound for all moduli except near-primes, saving $c^{-1/12}$ for products of two primes of the same size. Combining this with previous results for prime moduli, we achieve savings beyond the Pólya-Vinogradov range for all moduli. We give applications to moments of twisted cuspidal $L$-functions, and to large sieve inequalities for exceptional cusp forms with composite levels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08445 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-abelian amplification and bilinear forms with Kloosterman sums Pascadi, Alexandru Number Theory 11L05, 11T23, 20C15, 11M06, 11F11, 11F30 We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli $c$, using Fourier analysis on $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ and an amplification argument with non-abelian characters. For sums of length $\sqrt{c}$, our method produces a non-trivial bound for all moduli except near-primes, saving $c^{-1/12}$ for products of two primes of the same size. Combining this with previous results for prime moduli, we achieve savings beyond the Pólya-Vinogradov range for all moduli. We give applications to moments of twisted cuspidal $L$-functions, and to large sieve inequalities for exceptional cusp forms with composite levels. |
| title | Non-abelian amplification and bilinear forms with Kloosterman sums |
| topic | Number Theory 11L05, 11T23, 20C15, 11M06, 11F11, 11F30 |
| url | https://arxiv.org/abs/2511.08445 |