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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.02388 |
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| _version_ | 1866912742700482560 |
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| author | Wei, Bolun Yang, Liping |
| author_facet | Wei, Bolun Yang, Liping |
| contents | The infinity symmetric power $L$-functions play a fundamental role in Wan's groundbreaking work on Dwork's conjecture[16]. Building upon this foundation, Haessig[8] established the $p$-adic estimates for these $L$-functions in the case of the one-dimensional Kloosterman family. In this paper, we extend Haessig's results by deriving a uniform lower bound for the $q$-adic Newton polygon of the infinite symmetric power $L$-functions associated with the hyper-Kloosterman family. For the $1$-dimensional Kloosterman family, Haessig[8] showed that there is a $p$-adic cohomology theory for the infinity symmetric power $L$-function. In this paper, we prove there is also a cohomological description of the infinity symmetric power $L$-function for the hyper-Kloosterman family. By applying the Frobenius endomorphism to this cohomology, we derive a uniform lower bound for the corresponding $L$-function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02388 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite symmetric power L-functions of the hyper-Kloosterman family Wei, Bolun Yang, Liping Number Theory The infinity symmetric power $L$-functions play a fundamental role in Wan's groundbreaking work on Dwork's conjecture[16]. Building upon this foundation, Haessig[8] established the $p$-adic estimates for these $L$-functions in the case of the one-dimensional Kloosterman family. In this paper, we extend Haessig's results by deriving a uniform lower bound for the $q$-adic Newton polygon of the infinite symmetric power $L$-functions associated with the hyper-Kloosterman family. For the $1$-dimensional Kloosterman family, Haessig[8] showed that there is a $p$-adic cohomology theory for the infinity symmetric power $L$-function. In this paper, we prove there is also a cohomological description of the infinity symmetric power $L$-function for the hyper-Kloosterman family. By applying the Frobenius endomorphism to this cohomology, we derive a uniform lower bound for the corresponding $L$-function. |
| title | Infinite symmetric power L-functions of the hyper-Kloosterman family |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.02388 |