Algebraicity of the near central non-critical value of symmetric fourth $L$-functions 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 $\mathit{\Pi}$.