Algebraicity of ratios of special $L$-values for $\mathrm{GL}(n)$

Abstract

We prove, under certain assumptions, algebraicity of the ratio $L(m,\Pi \times \chi )/L(m,\Pi \times {\chi }^{\prime })$, where $\Pi$ is a cuspidal automorphic cohomological unitary representation of ${\mathrm{GL}}_n({\mathbb{A}}_{\mathbb{Q}})$, and $\chi$, ${\chi }^{\prime }$ are finite order Hecke characters such that ${\chi }_{\infty }={\chi }_{\infty }^{\prime }={\mathrm{sgn}}^r$, and $m,r$ are specific positive integers which depends only on ${\Pi }_{\infty }$. The methods in this article are a generalization of those in the work of Mahnkopf [Cohomology of arithmetic groups, parabolic subgroups and the special values of $L$-functions of GL(n), J. Inst. Math. Jussieu, 4 (2005)].