Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraicity and the $p$-adic Interpolation of Special $L$-values for certain Classical Groups

Yubo Jin

TL;DR

The paper develops a comprehensive doubling-method framework to represent standard L-functions of classical groups via global zeta integrals, explicitly constructing local sections to capture ramified factors. It proves algebraicity of certain special L-values after suitable normalization and constructs p-adic L-functions by interpolating these values across p-adic families, under ordinarity assumptions. The work extends Shimura/Deligne-type algebraicity results to symplectic, unitary, quaternionic unitary, and quaternionic orthogonal groups, and provides explicit local ramified L-factors and Fourier expansions of Eisenstein series to enable p-adic interpolation. The results have significant implications for arithmetic of automorphic L-values, p-adic interpolation, and connections to Shimura-variety frameworks and CM-periods.

Abstract

In this paper, we calculate the ramified local integrals in the doubling method and present an integral representation of standard $L$-functions for classical groups. We explicitly construct local sections of Eisenstein series such that the local ramified integrals represent certain ramified $L$-factors. As an application, we prove algebraicity of special $L$-values and construct $p$-adic $L$-functions for symplectic, unitary, quaternionic unitary and quaternionic orthogonal groups.

Algebraicity and the $p$-adic Interpolation of Special $L$-values for certain Classical Groups

TL;DR

The paper develops a comprehensive doubling-method framework to represent standard L-functions of classical groups via global zeta integrals, explicitly constructing local sections to capture ramified factors. It proves algebraicity of certain special L-values after suitable normalization and constructs p-adic L-functions by interpolating these values across p-adic families, under ordinarity assumptions. The work extends Shimura/Deligne-type algebraicity results to symplectic, unitary, quaternionic unitary, and quaternionic orthogonal groups, and provides explicit local ramified L-factors and Fourier expansions of Eisenstein series to enable p-adic interpolation. The results have significant implications for arithmetic of automorphic L-values, p-adic interpolation, and connections to Shimura-variety frameworks and CM-periods.

Abstract

In this paper, we calculate the ramified local integrals in the doubling method and present an integral representation of standard -functions for classical groups. We explicitly construct local sections of Eisenstein series such that the local ramified integrals represent certain ramified -factors. As an application, we prove algebraicity of special -values and construct -adic -functions for symplectic, unitary, quaternionic unitary and quaternionic orthogonal groups.
Paper Structure (38 sections, 33 theorems, 338 equations)

This paper contains 38 sections, 33 theorems, 338 equations.

Table of Contents

  1. Introduction
  2. Integral representations
  3. The algebraic result
  4. The $p$-adic $L$-function
  5. The Integral Representation of the Standard $L$-function
  6. The doubling embedding
  7. The global zeta integral
  8. Main result on integral representations
  9. Hecke Operators and local $L$-factors
  10. Unramified local $L$-factors
  11. Ramified local $L$-factors
  12. The split case
  13. Unramified local $L$-factors
  14. Ramified local $L$-factors
  15. The global $L$-function
  16. ...and 23 more sections

Key Result

Theorem 1.1

(Theorem theorem 2.2, theorem 6.1) There is a choice of $f_s$ such that where $C$ is some nonzero constant depending on $s$, $\phi'$ is some simple translate of $\phi$ and $\mathcal{Z}_{\infty}(s;\phi_{\infty},f_s^{\infty})$ a nonzero constant depending on the choice of the archimedean section $f_s^{\infty}$. When the underlying symmetric space of $G$ is hermitian, and

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 62 more