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

Abstract

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

References in corpus (1)

• On Deligne's periods for tensor product motives