标题： 对Hilbert模形式的对称四次$L$-函数的近中心非临界值的代数性 显示英文标题

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

摘要： 设$\mathit{\Pi}$为在全实数域${\mathbb F}$上的群${\rm GL}_2(\mathbb{A}_{\mathbb F})$的一个上同调不可约的尖形式自守表示，中心特征为$\omega_{\mathit{\Pi}}$。在本文中，我们证明了对$\mathit{\Pi}$的对称四次$L$函数在由$\omega_{\mathit{\Pi}}^{-2}$扭曲后的接近中心的非临界值的代数性。 代数性以标准化新形式的Petersson范数和Gelbart-Jacquet提升的最高次数Whittaker周期表达，其中标准化新形式属于$\mathit{\Pi}$，而Gelbart-Jacquet提升属于${\rm Sym}^2\mathit{\Pi}$，对应于$\mathit{\Pi}$。 显示英文摘要

摘要： 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 $\mathit{\Pi}$.