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

Abstract

Let $\mathitΠ$ be a cohomological irreducible cuspidal automorphic representation of ${\rm GL}_2(\mathbb{A}_{\mathbb F})$ with central character $ω_{\mathitΠ}$ 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 $\mathitΠ$ twisted by $ω_{\mathitΠ}^{-2}$. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of $\mathitΠ$ and the top degree Whittaker period of the Gelbart-Jacquet lift ${\rm Sym}^2\mathitΠ$ of $\mathitΠ$.