On the arithmetic of Shalika models and the critical values of $L$-functions for ${\rm GL}(2n)$

Abstract

Let $\Pi$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\mathbb A)$ over a totally real number field $F$. Suppose that $\Pi$ has a Shalika model. We define a rational structure on the Shalika model of $\Pi_f$. Comparing it with a rational structure on a realization of $\Pi_f$ in cuspidal cohomology in top-degree, we define certain periods $\omega^{\epsilon}(\Pi_f)$. We describe the behaviour of such top-degree periods upon twisting $\Pi$ by algebraic Hecke characters $\chi$ of $F$. Then we prove an algebraicity result for all the critical values of the standard $L$-functions $L(s, \Pi \otimes \chi)$; here we use the work of B. Sun on the non-vanishing of a certain quantity attached to $\Pi_\infty$. As an application, we obtain new algebraicity results in the following cases: Firstly, for the symmetric cube $L$-functions attached to holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly, for Rankin-Selberg $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$; assuming Langlands Functoriality, this generalizes to Rankin-Selberg $L$-functions of ${\rm GL}_n \times {\rm GL}_{n-1}$. Thirdly, for the degree four $L$-functions for ${\rm GSp}_4$. Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.