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 ω ɛ (Πf). 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 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 GL3 × GL2; assuming Langlands Functoriality, this generalizes to Ranking–Selberg L-functions of GLn × GLn−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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Abstract. We compute the arithmetic L-invariants (of Greenberg–Benois) of twists of symmetric powers of p-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on p-adic families of automorphic forms over symplectic and unitary groups. Combining these families with some explicit plethysm in the representation theory of GL(2), we construct global Galois cohomology classes with coefficients in the symmetric powers and provide formulae for the L-invariants in terms of logarithmic derivatives of Hecke eigenvalues.

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