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.
Abstract. Let K be an imaginary quadratic field. Let Π and Π ′ be irreducible generic cohomological automorphic representation of GL(n)/K and GL(n − 1)/K, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if Π is cuspidal and the weights of Π and Π ′ are in a standard relative position, the critical values of the Rankin-Selberg product L(s, Π × Π ′ ) are essentially algebraic multiples of the product of the Whittaker periods of Π and Π ′ . We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal Π can be given a motivic interpretation, and can also be related to a critical value of the adjoint L-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin-Selberg and adjoint L-functions are compatible with Deligne's conjecture. 1. Introduction L-functions can be attached both to automorphic representations and to arithmetic objects such as Galois representations or motives, and one implication of the Langlands program is that L-functions of the second kind are examples of L-functions of the first kind. Very few results of arithmetic interest can be proved about the second kind of L-functions until they have been identified with automorphic L-functions. For example, there is an extraordinarily deep web of conjectures relating the values at integer points of arithmetic (motivic) L-functions to cohomological invariants of the corresponding geometric (motivic) objects. In practically all the instances 1 where these conjectures have been proved, automorphic methods have proved indispensable. At the same time, there is a growing number of results on special values of automorphic L-functions that make no direct reference to arithmetic. Instead, the special values are written as algebraic multiples of complex invariants defined by means of representation theory. Examples relevant to the present paper ′ ) of a cuspdial automorphic representation Π of GL n and a cuspidal or abelian automorphic representation Π ′ of GL n−1 , where the general linear groups are over an imaginary quadratic field K. Here, the notion "abelian automorphic" refers to a representation, which is a tempered Eisenstein representation induced from a Borel subgroup of GL n−1 . In view of Raghuram's recent preprint [39], which we received after writing a first version of this paper, the inclusion of such automorphic representations Π ′ is the new feature of this result. The theorem applies in particular when Π and Π ′ are obtained by base change from cohomological cuspidal representations π and π ′ of unitary groups. As in [35], [40] and [39] the critical values of these L-functions are then expressed in terms of Whittaker periods, which are p...
Let G be a connected, reductive algebraic group over a number field F and let E be an algebraic representation of G ∞ . In this paper we describe the Eisenstein cohomology H q Eis (G, E) of G below a certain degree q res in terms of Franke's filtration of the space of automorphic forms. This entails a description of the mapEis (G, E), q < q res , for all automorphic representations Π of G(A) appearing in the residual spectrum. Moreover, we show that below an easily computable degree q max , the space of Eisenstein cohomology H q Eis (G, E) is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of GL n and the split classical groups of type B n , C n , D n .
In this paper we investigate arithmetic properties of automorphic forms on the group G ′ = GLm/D, for a central division-algebra D over an arbitrary number field F . The results of this article are generalizations of results in the split case, i.e., D = F , by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke-Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G ′ to statements on automorphic forms of its split form using the global Jacquet-Langlands correspondence developed by Badulescu and Badulescu-Renard. Beside that we prove that the local version of the Jacquet-Langlands transfer at an archimedean place preserves the property of being cohomological.
Let F be a totally real number field and E/F a totally imaginary quadratic extension of F . Let Π be a cohomological, conjugate self-dual cuspidal automorphic representation of GL n (A E ). Under a certain non-vanishing condition we relate the residue and the value of the Asai L-functions at s = 1 with rational structures obtained from the cohomologies in top and bottom degrees via the Whittaker coefficient map. This generalizes a result in Eric Urban's thesis when n = 2, as well as a result of the first two named authors, both in the case F = Q.
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