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On the Siegel series in terms of lattice counting

Sungmun Cho, Taeyeoup Kang

Abstract

In this paper we describe each coefficient of the Siegel series associated to a quadratic $\mathfrak{o}$-lattice $L$ in terms of lattice counting problems, where $\mathfrak{o}$ is the ring of integers of a non-Archimedean local field of characteristic $0$. Under the restriction that $p$ is odd and that the dimension of the radical of the quadratic space $L\otimesκ$ on the residue field $κ$ is at most $2$, we provide explicit values of coefficients and reprove the functional equation of the Siegel series.

On the Siegel series in terms of lattice counting

Abstract

In this paper we describe each coefficient of the Siegel series associated to a quadratic -lattice in terms of lattice counting problems, where is the ring of integers of a non-Archimedean local field of characteristic . Under the restriction that is odd and that the dimension of the radical of the quadratic space on the residue field is at most , we provide explicit values of coefficients and reprove the functional equation of the Siegel series.
Paper Structure (7 sections, 14 theorems, 23 equations)

This paper contains 7 sections, 14 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Description of the Siegel series in CY
  4. Siegel series in terms of lattice counting
  5. Explicit values of coefficients when $p$ is odd and $n_0 \geq n-2$
  6. Formula for $\#\mathcal{S}_{(L,a^\pm,b)}$
  7. Explicit values of coefficients of the Siegel series

Key Result

Proposition 3.4

CY For a quadratic lattice $L$, let $n_0$ be the number of $0$'s in $\mathrm{GK}(L)=(a_1, \cdots, a_n)$. We have where Here, if $a$ is odd or $0$, then we ignore one of $\mathcal{S}_{(L, a^{+}, b)}$ or $\mathcal{S}_{(L, a^{-}, b)}$. If $a$ is even and positive, then we count the summands involving $\mathcal{S}_{(L, a^{+}, b)}$ and $\mathcal{S}_{(L, a^{-}, b)}$ separately.

Theorems & Definitions (25)

  • Remark 1.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Proposition 3.4
  • Definition 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Corollary 4.4
  • proof
  • ...and 15 more