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Zeta functions and nonvanishing theorems for toric periods on $\mathrm{GL}_2$

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 cahracter. We study the zeta functions with period integrals on $π$ for the perhomogeneous vector space $(D^\times\times D^\times\times\mathrm{GL}_2, D\oplus D)$. We show their meromorphic continuation and functional equation, determine the location and orders of possible poles and compute the residue. Arguing along the theory of Saito and computing unramified local factors, the explicit formula of the zeta functions is obtained. Counting the order of possible poles of this explicit formula, we show that if $L(1/2, π)\neq0$, there are infinitely many quadratic extension $E$ of $F$ which embeds in $D$, such that $π$ has nonvanishing toric period with respect to $E$.

Zeta functions and nonvanishing theorems for toric periods on $\mathrm{GL}_2$

Abstract

Let be a number field and a quaternion algebra over . Take a cuspidal automorphic representation of with trivial central cahracter. We study the zeta functions with period integrals on for the perhomogeneous vector space . We show their meromorphic continuation and functional equation, determine the location and orders of possible poles and compute the residue. Arguing along the theory of Saito and computing unramified local factors, the explicit formula of the zeta functions is obtained. Counting the order of possible poles of this explicit formula, we show that if , there are infinitely many quadratic extension of which embeds in , such that has nonvanishing toric period with respect to .

Paper Structure

This paper contains 47 sections, 41 theorems, 316 equations.

Table of Contents

  1. Introduction and main results
  2. Global zeta functions
  3. Toric periods and local zeta functions
  4. Explicit formula
  5. Nonvanishing theorems
  6. Functional equations
  7. Prior studies on zeta functions
  8. Organization of this paper
  9. Preliminaries
  10. Quaternion algebra
  11. The space of pairs of quaternion algebras
  12. Remarks
  13. Orbit decompositions
  14. Global zeta functions with automorphic forms
  15. Number fields
  16. ...and 32 more sections

Key Result

Theorem 1.2

Assume that $S$ satisfies Condition condition. Take a Schwartz function $\Phi=\otimes_{v\in\Sigma} \Phi_v$ on $V(\mathbb{A})$ such that $\Phi_v$ is the characteristic function of $V(\mathfrak{o}_v)$ for every $v\notin S$. For sufficiently large ${\mathop{\mathrm{Re}}}(s)>0$, we obtain where $\mathcal{D}_E^S(\pi,s)\coloneqq \prod_{v\notin S}\mathcal{D}_{E_v}(\pi_v,s)$, and

Theorems & Definitions (75)

  • Theorem 1.2
  • proof
  • Theorem 1.3
  • proof
  • Theorem 1.4
  • proof
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Proposition 1.8
  • ...and 65 more