On the arithmetic of Shalika models and the critical values of L-functions for GL_2n

Abstract

Let Π be a cohomological cuspidal automorphic representation of GL_2n(A) over a totally real number eld F. Suppose that Π has a Shalika model. We define a rational structure on the Shalika model of Π_f. Comparing it with a rational structure on a realization of Π_f in cuspidal cohomology in top-degree, we define certain periods ω^ϵ(Π_f). We describe the behaviour 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 GL_n × GL_n-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 de ned by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.