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

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

2011-06-21

arxiv.org/abs/1106.4249v2

Mathematics

Number Theory

This is a substantially revised version, taking into account some recent developments, which make our results unconditional: T

Scientific paper

Let \Pi be a cohomological cuspidal automorphic representation of GL_2n(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 recent 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 GL_3 \times GL_2; assuming Langlands Functoriality, this generalizes to Ranking-Selberg L-functions of GL_n \times GL_{n-1}. Thirdly, for the degree four L-functions for 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.

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