Counting Lattices with Local Hecke Series

TL;DR

This paper develops a method to count maximal lattices over p-adic and rational fields using Hecke series, leading to new and existing zeta functions for classical groups, simplifying previous approaches.

Contribution

It introduces a novel approach to counting lattices via Hecke series, deriving new zeta functions for orthogonal groups and unifying previous results.

Findings

Derived zeta functions for classical groups

Obtained new zeta functions for orthogonal groups

Unified counting methods with existing zeta functions

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.