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Central Values of Degree Six L-functions: The Case of Hilbert Modular Forms

Utkarsh Agrawal

TL;DR

The paper derives a precise formula for the central value $L(\kappa_0+\kappa'_0, \mathrm{Sym}^2 g\times f)$ of the completed $L$-function for Hilbert modular forms over a totally real field $F$ by explicitly evaluating local period integrals in the refined Gan–Gross–Prasad setting for $\mathrm{SL}_2\times\widetilde{\mathrm{SL}}_2$. The authors express the central value as a squared automorphic period involving a nearly holomorphic Jacobi form $F_{h_{ub}}$ acted on by a differential operator $\Delta^{(r)}$, with normalization by Petersson inner products and the completed zeta factor $\xi_F(2)$; archimedean and nonarchimedean local components are computed in detail, and the constant factors are made explicit. They establish rationality results in two extremal configurations (purely balanced or purely unbalanced) using Shimura’s integral-weight and half-integral-weight theories, and they formulate a Deligne-compatible conjecture for the general case. The work links central $L$-values to explicit automorphic periods and local data, enabling algebraicity results and suggesting a concrete pathway to Deligne-type period computations in this higher-rank setting. Overall, the paper provides a concrete, computable bridge between central $L$-values for Hilbert modular forms and explicit automorphic periods, with potential applications to arithmetic and motivic questions tied to Jacobi and Waldspurger-type correspondences.

Abstract

In this paper we give a formula for the central value of the completed $L$-function $L(s,Sym^{2} g\times f)$, where $f$ and $g$ are Hilbert newforms, by explicitly computing the local integrals appearing in the refined Gan-Gross-Prasad conjecture for $SL_{2}\times\tilde{SL_{2}}$. We also work out the rationality of this value in some special cases and give a conjecture for the general case.

Central Values of Degree Six L-functions: The Case of Hilbert Modular Forms

TL;DR

The paper derives a precise formula for the central value of the completed -function for Hilbert modular forms over a totally real field by explicitly evaluating local period integrals in the refined Gan–Gross–Prasad setting for . The authors express the central value as a squared automorphic period involving a nearly holomorphic Jacobi form acted on by a differential operator , with normalization by Petersson inner products and the completed zeta factor ; archimedean and nonarchimedean local components are computed in detail, and the constant factors are made explicit. They establish rationality results in two extremal configurations (purely balanced or purely unbalanced) using Shimura’s integral-weight and half-integral-weight theories, and they formulate a Deligne-compatible conjecture for the general case. The work links central -values to explicit automorphic periods and local data, enabling algebraicity results and suggesting a concrete pathway to Deligne-type period computations in this higher-rank setting. Overall, the paper provides a concrete, computable bridge between central -values for Hilbert modular forms and explicit automorphic periods, with potential applications to arithmetic and motivic questions tied to Jacobi and Waldspurger-type correspondences.

Abstract

In this paper we give a formula for the central value of the completed -function , where and are Hilbert newforms, by explicitly computing the local integrals appearing in the refined Gan-Gross-Prasad conjecture for . We also work out the rationality of this value in some special cases and give a conjecture for the general case.

Paper Structure

This paper contains 27 sections, 28 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. The base field
  4. Sum over a subset
  5. The sets
  6. The completed Dedekind zeta function
  7. Measures
  8. The Jacobi group $\mathop{\mathrm{J}}\nolimits$
  9. Lie algebra of the Jacobi group
  10. The Schrödinger-Weil representation $\omega_{\psi}$
  11. The Waldspurger packet $\mathop{\mathrm{Wd}}\nolimits_{\psi}(\pi)$
  12. The L-function $L(s, \mathop{\mathrm{Sym}}\nolimits^{2}\tau\times \pi)$
  13. Nearly holomorphic automorphic forms
  14. The Main Formula
  15. Local Components
  16. ...and 12 more sections

Key Result

Theorem 1

There is a nonzero scalar $c_{\kappa,\kappa'}$, such that where $\xi_{F}$ is the completed Dedekind zeta function of $F$ and $D_{F}$ is the discriminant of $F$.

Theorems & Definitions (65)

  • Theorem
  • Remark
  • Remark
  • Theorem
  • Remark
  • Remark
  • Remark
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • ...and 55 more