On the arithmetic of Shalika models and the critical values of L -functions for GL 2 n

Let Π be a cohomological cuspidal automorphic representation of GL₂ₙ(A) over a totally real number field F. Suppose that Π has a Shalika model. We define a rational structure on the Shalika model of Πf. Comparing it with a rational structure on a realization of Πf in cuspidal cohomology in top-degree, we define certain periods ωε(Πf). We describe the behavior of such top-degree periods upon twisting Π by algebraic Hecke characters χ of F. Then we prove an algebraicity result for all the critical values of the standard L-functions L(s,Π⊗χ); here we use the recent work of B. Sun on the non-vanishing of a certain quantity attached to Π_∞. As applications, we obtain 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 certain (self-dual of symplectic type) Rankin-Selberg L-functions for GL₃× GL₂; assuming Langlands Functoriality, this generalizes to certain Rankin-Selberg L-functions of GLₙ× GLₙ₋₁. Thirdly, for the degree four L-functions attached to Siegel modular forms of genus $2$ and full level. 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.