On the Arithmetic of Shalika Models and the Critical Values of L-Functions for GL(2n)

Abstract

Let II be a cohomological cuspidal automorphic representation of GL(2n), (A) over a totally real number field F. Suppose that II has a Shalika model. We define a rational structure on the Shalika model of IIf. Comparing it with a rational structure on a realization of IIf in cuspidal cohomology in top-degree, we define certain periods omega(is an element of)(IIf). We describe the behavior of such top-degree periods upon twisting II by algebraic Hecke characters x of F. Then we prove an algebraicity result for all the critical values of the standard L-functions L(s,II circle times chi); here we use the recent work of B. Sun on the non-vanishing of a certain quantity attached to II. 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 GL(3) x GL(2); assuming Langlands Functoriality, this generalizes to certain Rankin-Selberg L-functions of GL(n), x 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 defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.