Abstract. We prove an algebraicity result for the central critical value of certain RankinSelberg L-functions for GLn ×GLn−1. This is a generalization and refinement of the results of Harder [15], Kazhdan, Mazur and Schmidt [23], Mahnkopf [29], and Kasten and Schmidt [22]. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands' functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. We comment on the compatibility of our results with Deligne's conjecture on the critical values of motivic L-functions. These results, as in the above mentioned works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.

Let π be a regular algebraic cuspidal automorphic representation of GL n (A F ) for a number field F . We consider certain periods attached to π. These periods were originally defined by Harder when n = 2, and later by Mahnkopf when F = Q. In the first part of the paper we analyze the behaviour of these periods upon twisting π by algebraic Hecke characters. In the latter part of the paper we consider Shimura's periods associated to a modular form. If ϕ χ is the cusp form associated to a character χ of a quadratic extension, then we relate the periods of ϕ χ n to those of ϕ χ , and as a consequence give another proof of Deligne's conjecture on the critical values of symmetric power L-functions associated to dihedral modular forms. Finally, we make some remarks on the symmetric fourth power L-functions.

Abstract. Let Π be a cohomological cuspidal automorphic representation of GL 2n (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 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 ǫ (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 GL 3 × GL 2 ; assuming Langlands Functoriality, this generalizes to certain Rankin-Selberg L-functions of GLn × 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.

In a previous article we had proved an algebraicity result for the central critical value for

Abstract. This is an announcement of results on rank-one Eisenstein cohomology of GL N , with N ≥ 3 an odd integer, and algebraicity theorems for ratios of successive critical values of certain Rankin-Selberg L-functions for GLn × GL n ′ when n is even and n ′ is odd. Résumé: Cette note est une annonce de résultats sur la cohomologie d'Eisenstein de rang un de GL N , avec N ≥ 3 un entier impair, et de théorèmes d'algébricité pour les rapports de valeurs critiques successives de certaines fonctions L de Rankin-Selberg pour GLn × GL n ′ lorsque n est pair et n ′ est impair.