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

Let $\mathrm{\Pi }$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\mathbb A)$overatotallyrealnumberfield$F$\mathrm{.}Supposethat$\Pi$hasaShalikamodel\mathrm{.}WedefinearationalstructureontheShalikamodelof$\Pi_f$\mathrm{.}Comparingitwitharationalstructureonarealizationof$\Pi_f$ in cuspidal cohomology in top-degree, we define certain periods ${\omega }^{ϵ}({\mathrm{\Pi }}_f)$. We describe the behaviour of such top-degree periods upon twisting $\mathrm{\Pi }$ by algebraic Hecke characters $\chi$ of $F$. Then we prove an algebraicity result for all the critical values of the standard $L$-functions $L(s,\mathrm{\Pi }\otimes \chi )$; here we use the work of B. Sun on the non-vanishing of a certain quantity attached to ${\mathrm{\Pi }}_{\mathrm{\infty }}$. 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 ${\mathrm{G}\mathrm{L}}_3\times {\mathrm{G}\mathrm{L}}_2$; assuming Langlands Functoriality, this generalizes to Rankin-Selberg $L$-functions of ${\mathrm{G}\mathrm{L}}_n\times {\mathrm{G}\mathrm{L}}_{n-1}$. Thirdly, for the degree four $L$-functions for ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$. 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.