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Explicit mean value theorems for toric periods and automorphic $L$-functions

Miyu Suzuki, Satoshi Wakatsuki

Abstract

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $π$ of $D_{\mathbb{A}}^\times$ with trivial central character and a cusp form $φ$ in $π$. Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of $φ$ with respect to quadratic algebras over $F$. The result can also be written as a mean value formula for the central values of automorphic $L$-functions twisted by quadratic characters.

Explicit mean value theorems for toric periods and automorphic $L$-functions

Abstract

Let be a number field and a quaternion algebra over . Take a cuspidal automorphic representation of with trivial central character and a cusp form in . Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of with respect to quadratic algebras over . The result can also be written as a mean value formula for the central values of automorphic -functions twisted by quadratic characters.

Paper Structure

This paper contains 24 sections, 29 theorems, 200 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Methods and related works
  4. Organization of the paper
  5. Mean value theorems
  6. Preliminaries
  7. Number fields
  8. Quaternion algebras
  9. Measures on algebraic groups
  10. Toric periods
  11. Mean value theorem for toric periods
  12. Mean value theorem for elliptic modular forms
  13. Mean value theorem for algebraic modular forms
  14. Prehomogeneous zeta function
  15. Definition and basic properties
  16. ...and 9 more sections

Key Result

Theorem 1

The limit exists, where the sum is over $E\in X(D, \mathcal{E}_S)$ such that $N({\mathfrak{f}}_E^S)<x$. The value of the above limit equals

Theorems & Definitions (55)

  • Theorem 1: A special case of Theorem \ref{['thm:main']}
  • Theorem 2: A special case of Corollary \ref{['cor:mvfL']}
  • Theorem 3: Corollary \ref{['cor:mvfhol']}
  • Theorem 5
  • Remark 6
  • Remark 7
  • Corollary 8
  • Remark 9
  • Theorem 10
  • proof
  • ...and 45 more