Counting Lattices with Local Hecke Series
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914227908771840 |
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| author | Bhowmik, Gautami Tsuzuki, Masao |
| author_facet | Bhowmik, Gautami Tsuzuki, Masao |
| contents | We count the maximal lattices over $p$-adic fields and the rational number field. For this, we use the theory of Hecke series for a reductive group over nonarchimedean local fields, which was developed by Andrianov and Hina-Sugano. By treating the Euler factors of the counting Dirichlet series for lattices, we obtain zeta functions of classical groups, which were earlier studied with $p$-adic cone integrals. When our counting series equals the existing zeta functions of groups, we recover the known results in a simple way. Further we obtain some new zeta functions for non-split even orthogonal and odd orthogonal groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting Lattices with Local Hecke Series Bhowmik, Gautami Tsuzuki, Masao Number Theory primary 11M41, secondary 11E45, 11E57, 11N45, 20D30 We count the maximal lattices over $p$-adic fields and the rational number field. For this, we use the theory of Hecke series for a reductive group over nonarchimedean local fields, which was developed by Andrianov and Hina-Sugano. By treating the Euler factors of the counting Dirichlet series for lattices, we obtain zeta functions of classical groups, which were earlier studied with $p$-adic cone integrals. When our counting series equals the existing zeta functions of groups, we recover the known results in a simple way. Further we obtain some new zeta functions for non-split even orthogonal and odd orthogonal groups. |
| title | Counting Lattices with Local Hecke Series |
| topic | Number Theory primary 11M41, secondary 11E45, 11E57, 11N45, 20D30 |
| url | https://arxiv.org/abs/2512.24690 |