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

Let Π be a cohomological irreducible cuspidal automorphic representation of GL2(AF) with central character ωΠ over a totally real number field F. In this paper, we prove the algebraicity of the near central non-critical value of the symmetric fourth L-function of Π twisted by ωΠ−2. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of Π and the top degree Whittaker period of the Gelbart–Jacquet lift Sym2Π of Π.