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Adelic Rogers integral formula

Seungki Kim

Abstract

We formulate and prove the extension of the Rogers integral formula to the adeles of number fields. We also prove the second moment formulas for a few important cases, enabling a number of classical and recent applications of the formula to extend immediately to any number field.

Adelic Rogers integral formula

Abstract

We formulate and prove the extension of the Rogers integral formula to the adeles of number fields. We also prove the second moment formulas for a few important cases, enabling a number of classical and recent applications of the formula to extend immediately to any number field.
Paper Structure (27 sections, 33 theorems, 236 equations)

This paper contains 27 sections, 33 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. History and motivation
  3. The main theorem
  4. Second moment estimates
  5. Organization
  6. Preliminaries
  7. Number fields
  8. Adeles
  9. Tamagawa measure
  10. Height
  11. Hecke operators
  12. Miscellaneous
  13. The "level 1" case
  14. A reduction, and a word about the proof
  15. Proof of Proposition \ref{['prop:main']}: preparation
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $k < n$ be positive integers, and $f:(\mathbb{R}^n)^k \rightarrow \mathbb{R}$ be a Borel integrable function. Let $X_n = \mathrm{SL}(n,\mathbb{Z}) \backslash \mathrm{SL}(n,\mathbb{R})$ be the moduli space of the lattices in $\mathbb{R}^n$ of covolume $1$, and $\mu$ be the unique right $\mathrm{S In addition, let us interpret $(\mathbb{R}^n)^k = \mathrm{Mat}_{k \times n}(\mathbb{R})$, the set o

Theorems & Definitions (59)

  • Theorem 1.1: Rogers Rog55, Siegel Sie45 for case $k=1$
  • Theorem 1.2
  • Corollary
  • Theorem 1.3
  • Corollary
  • Remark
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 49 more