On the arithmetic of Shalika models and the critical values of L-functions for GL2n

Grobner, Harald ; Raghuram, A. ; Gan, Wee Teck (2014) On the arithmetic of Shalika models and the critical values of L-functions for GL2n American Journal of Mathematics, 136 (3). pp. 675-728. ISSN 0002-9327

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Official URL: http://muse.jhu.edu/article/546015/summary

Related URL: http://dx.doi.org/10.1353/ajm.2014.0021


Let Π be a cohomological cuspidal automorphic representation of GL2n(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 Πf. Comparing it with a rational structure on a realization of Πf in cuspidal cohomology in top-degree, we define certain periods ωϵf). 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 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 de ned by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.

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