Modular invariants for real quadratic fields and Kloosterman sums
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arXiv
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| Format: | Preprint |
| Published: |
2018
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| _version_ | 1866912074736599040 |
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| author | Andersen, Nickolas Duke, William |
| author_facet | Andersen, Nickolas Duke, William |
| contents | We investigate the asymptotic distribution of integrals of the $j$-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight that is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov's formula where the spectral data is restricted to half-integral weight forms in the Kohnen plus space, and we apply Young's hybrid subconvexity estimates for twisted modular $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1801_08174 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Modular invariants for real quadratic fields and Kloosterman sums Andersen, Nickolas Duke, William Number Theory We investigate the asymptotic distribution of integrals of the $j$-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight that is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov's formula where the spectral data is restricted to half-integral weight forms in the Kohnen plus space, and we apply Young's hybrid subconvexity estimates for twisted modular $L$-functions. |
| title | Modular invariants for real quadratic fields and Kloosterman sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/1801.08174 |