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arXiv · 2512.24690

Counting Lattices with Local Hecke Series

Gautami Bhowmik·Masao Tsuzuki
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Abstract

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.

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Gautami Bhowmik, Masao Tsuzuki. 2025-12-31. Counting Lattices with Local Hecke Series. https://arxiv.org/abs/2512.24690

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