Algebraicity of the near central non-critical value of symmetric fourth $L$-functions for Hilbert modular forms

TL;DR

This paper proves the algebraicity of a specific near central value of the symmetric fourth L-function for Hilbert modular forms, relating it to Petersson norms and Whittaker periods.

Contribution

It establishes the algebraicity of a non-critical L-value for symmetric fourth power L-functions of Hilbert modular forms, linking it to automorphic periods.

Findings

Proves algebraicity of the near central non-critical value of the symmetric fourth L-function.

Expresses algebraicity in terms of Petersson norm and Whittaker period.

Connects special L-values with automorphic periods for Hilbert modular forms.

Abstract

Let $\mathit{\Pi}$ be a cohomological irreducible cuspidal automorphic representation of ${\rm GL}_2(\mathbb{A}_{\mathbb F})$ with central character $\omega_{\mathit{\Pi}}$ over a totally real number field ${\mathbb F}$. In this paper, we prove the algebraicity of the near central non-critical value of the symmetric fourth $L$-function of $\mathit{\Pi}$ twisted by $\omega_{\mathit{\Pi}}^{-2}$. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of $\mathit{\Pi}$ and the top degree Whittaker period of the Gelbart-Jacquet lift ${\rm Sym}^2\mathit{\Pi}$ of $\mathit{\Pi}$.