Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On $p$-adic Measures for Quaternionic Modular Forms

Yubo Jin

TL;DR

The paper investigates the special values of standard $L$-functions for quaternionic modular forms via the doubling method, obtaining an integral representation through pullbacks of Siegel Eisenstein series. It then constructs a $p$-adic measure that interpolates these special values, under $p$-ordinary hypotheses and a $p$-stabilized eigenform, by combining adelic Eisenstein machinery with carefully designed differential operators to control weight and level. The authors provide explicit interpolation formulas for the $p$-adic measure in two cases (Case I and Case II), including precise gamma factors, Gauss sums, and modified $p$-Euler factors, and establish algebraicity and Galois-equivariance properties of the special values via Fourier-expansion techniques. They also discuss the current limitations of $p$-adic theory for quaternionic Shimura varieties and outline directions toward extending $p$-adic L-functions for ordinary families in this setting, highlighting the potential for broader arithmetic applications.

Abstract

The purpose of this paper is to study the special values of the standard $L$-functions for quaternionic modular forms using the doubling method. We obtain an integral representation for the $L$-function twisted by a character and construct the $p$-adic measure interpolating certain special $L$-values.

On $p$-adic Measures for Quaternionic Modular Forms

TL;DR

The paper investigates the special values of standard -functions for quaternionic modular forms via the doubling method, obtaining an integral representation through pullbacks of Siegel Eisenstein series. It then constructs a -adic measure that interpolates these special values, under -ordinary hypotheses and a -stabilized eigenform, by combining adelic Eisenstein machinery with carefully designed differential operators to control weight and level. The authors provide explicit interpolation formulas for the -adic measure in two cases (Case I and Case II), including precise gamma factors, Gauss sums, and modified -Euler factors, and establish algebraicity and Galois-equivariance properties of the special values via Fourier-expansion techniques. They also discuss the current limitations of -adic theory for quaternionic Shimura varieties and outline directions toward extending -adic L-functions for ordinary families in this setting, highlighting the potential for broader arithmetic applications.

Abstract

The purpose of this paper is to study the special values of the standard -functions for quaternionic modular forms using the doubling method. We obtain an integral representation for the -function twisted by a character and construct the -adic measure interpolating certain special -values.
Paper Structure (19 sections, 25 theorems, 251 equations)

This paper contains 19 sections, 25 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Quaternionic Modular Forms and $L$-functions
  3. Quaternion algebra and algebraic groups
  4. Symmetric spaces
  5. Definition of modular forms
  6. Hecke operators and $L$-functions
  7. $p$-stabilisation
  8. Eisenstein Series and the Doubling Method
  9. Siegel Eisenstein series and its Fourier expansion
  10. The integral representation: nontrivial twist
  11. The modification at $p$
  12. Differential Operators and Classical Reformulations
  13. Preliminaries
  14. Construction of differential operators
  15. Differential operators
  16. ...and 4 more sections

Key Result

Theorem 1.1

(Theorem 6.4,6.5) There is a unique $p$-adic measure $\mu$ on $\mathbb{Z}_p^{\times}$ such that: (1) (Case I) For $t=2n+1,...,\kappa$ and primitive Dirichlet character $\chi$ of $p$-power conductor $c_{\chi}=p^{\mathfrak{c}}$ we have (2) (Case II) For $t=2n+1,...,\kappa$ and primitive Dirichlet character $\chi$ of $p$-power conductor $c_{\chi}=p^{\mathfrak{c}}$ with $\chi(-1)=(-1)^t$ we have Her

Theorems & Definitions (45)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 35 more