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

TL;DR

This paper derives a formula for the central value of certain degree six L-functions associated with Hilbert modular forms, explicitly computing local integrals and exploring rationality properties in special cases.

Contribution

It provides the first explicit formula for the central values of degree six L-functions for Hilbert modular forms, advancing understanding of their special values and rationality.

Findings

Explicit formula for $L(1/2,Sym^{2} g imes f)$ in terms of local integrals

Verification of rationality of the central value in specific cases

Conjecture on the general rationality of these L-values

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.