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

Abstract

Let Π be a cohomological irreducible cuspidal automorphic representation of GL 2 (A F ) with central character ω Π 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 ω −2 Π . 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 Sym 2 Π of Π . Contents 1. Introduction 1 2. Whittaker periods for GL 3 5 3. Proof of Main Result 12 4. Algebraicity of the Rankin-Selberg L-functions for GL 3 × GL 2 13 5. Algebraicity of the adjoint L-functions for GL 3 23 References 27