Short sums of trace functions over function fields and their applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915700740718592 |
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| author | Sawin, Will Shusterman, Mark |
| author_facet | Sawin, Will Shusterman, Mark |
| contents | For large enough (but fixed) prime powers $q$, and trace functions to squarefree moduli in $\mathbb{F}_q[u]$ with slopes at most $1$ at infinity, and no Artin--Schreier factors in their geometric global monodromy, we come close to square-root cancellation in short sums. A special case is a function field version of Hooley's Hypothesis $R^*$ for short Kloosterman sums. As a result, we are able to make progress on several problems in analytic number theory over $\mathbb{F}_q[u]$ such as Mordell's problem on the least residue class not represented by a polynomial and the variance of short Kloosterman sums. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Short sums of trace functions over function fields and their applications Sawin, Will Shusterman, Mark Number Theory Algebraic Geometry 11L07 11L40 14F20 11T06 11L05 For large enough (but fixed) prime powers $q$, and trace functions to squarefree moduli in $\mathbb{F}_q[u]$ with slopes at most $1$ at infinity, and no Artin--Schreier factors in their geometric global monodromy, we come close to square-root cancellation in short sums. A special case is a function field version of Hooley's Hypothesis $R^*$ for short Kloosterman sums. As a result, we are able to make progress on several problems in analytic number theory over $\mathbb{F}_q[u]$ such as Mordell's problem on the least residue class not represented by a polynomial and the variance of short Kloosterman sums. |
| title | Short sums of trace functions over function fields and their applications |
| topic | Number Theory Algebraic Geometry 11L07 11L40 14F20 11T06 11L05 |
| url | https://arxiv.org/abs/2512.24080 |