1. Algebraicity of the near central non-critical values of symmetric fourth L-functions for Hilbert modular forms
- Author
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Shih-Yu Chen
- Subjects
Statistics::Theory ,Pure mathematics ,Algebra and Number Theory ,Non critical ,Mathematics - Number Theory ,Statistics::Applications ,Degree (graph theory) ,Mathematics::Number Theory ,Modular form ,Lift (mathematics) ,Character (mathematics) ,Norm (mathematics) ,FOS: Mathematics ,Number Theory (math.NT) ,Totally real number field ,Mathematics::Representation Theory ,Representation (mathematics) ,Mathematics - 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 $\mathit{\Pi}$.
- Published
- 2022
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