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Counting Lattices with Local Hecke Series

Gautami Bhowmik, Masao Tsuzuki

TL;DR

This work develops a unified, Hecke-series–driven framework to count maximal lattices over $p$-adic fields and over $\mathbb{Q}$, recasting lattice enumeration as a zeta function with Euler factors governed by ASH (Andrianov–Hina–Sugano) polynomials. It provides explicit, uniform formulas for the local and global counting zetas in the ${\rm C}_{\ell}$ and ${\rm D}_{\ell}$ cases, and extends to the ${\rm B}_{\ell}$ case and non-split orthogonal groups, revealing new zeta functions and asymptotics. The analysis combines local index-series theory, explicit numerator polynomials, and Euler-product globalization, tying the results to group zeta functions studied by du Sautoy–Woodward and Igusa-type p-adic zeta integrals. The findings yield precise pole structures and Tauberian asymptotics for lattice counts, and open avenues for further exploration of hermitian maximal lattices and related forms. Overall, the paper advances a uniform, algebraic-analytic method for lattice counting with broad implications for automorphic and zeta-function theory.

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.

Counting Lattices with Local Hecke Series

TL;DR

This work develops a unified, Hecke-series–driven framework to count maximal lattices over -adic fields and over , recasting lattice enumeration as a zeta function with Euler factors governed by ASH (Andrianov–Hina–Sugano) polynomials. It provides explicit, uniform formulas for the local and global counting zetas in the and cases, and extends to the case and non-split orthogonal groups, revealing new zeta functions and asymptotics. The analysis combines local index-series theory, explicit numerator polynomials, and Euler-product globalization, tying the results to group zeta functions studied by du Sautoy–Woodward and Igusa-type p-adic zeta integrals. The findings yield precise pole structures and Tauberian asymptotics for lattice counts, and open avenues for further exploration of hermitian maximal lattices and related forms. Overall, the paper advances a uniform, algebraic-analytic method for lattice counting with broad implications for automorphic and zeta-function theory.

Abstract

We count the maximal lattices over -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 -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.
Paper Structure (26 sections, 56 theorems, 211 equations)

This paper contains 26 sections, 56 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Counting problem of the number of maximal lattices
  3. Description of main results
  4. Hecke series
  5. Detailed organization of paper
  6. The -case: the local theory
  7. Lattices in a quadratic space over p-adic fields
  8. Maximal lattices
  9. The local index series
  10. The numerator of the index function series: the case
  11. The properties of
  12. The numerator of the index function series: the case
  13. Proof of Theorem \ref{['IdxFtSrf2-T1']}: step 1
  14. Proof of Theorem \ref{['IdxFtSrf2-T1']}: step 2
  15. Completion of proof
  16. ...and 11 more sections

Key Result

Theorem 1.8

The ASH polynomial satisfies the functional equation For $j=d\alpha_{\ell,d}^{(\varepsilon)}$ and for $j=d\beta_{\ell,d}^{(\varepsilon)}$, we have that $\Gamma_{\ell,d}^{(\varepsilon)}(j)=1$

Theorems & Definitions (120)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 110 more