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We prove explicit rationality-results for Asai- $L$-functions, $L^S(s,\Pi',{\rm As}^\pm)$, and Rankin-Selberg $L$-functions, $L^S(s,\Pi\times\Pi')$, over arbitrary CM-fields $F$, relating critical values to explicit powers of $(2\pi i)$. Besides determining the contribution of archimedean zeta-integrals to our formulas as concrete powers of $(2\pi i)$, it is one of the advantages of our approach, that it applies to very general non-cuspidal isobaric automorphic representations $\Pi'$ of ${\rm GL}_n(\mathbb A_F)$. As an application, this enables us to establish a certain algebraic version of the Gan--Gross--Prasad conjecture, as refined by N.\ Harris, for totally definite unitary groups. As another application we obtain a generalization of a result of Harder--Raghuram on quotients of consecutive critical values, proved by them for totally real fields, and achieved here for arbitrary CM-fields $F$ and pairs $(\Pi,\Pi')$ of relative rank one.

Let $$G=\mathrm{GL}_{2n}$$ over a totally real number field F and $$n\ge 2$$ . Let $$\Pi $$ be a cuspidal automorphic representation of $$G(\mathbb {A})$$ , which is cohomological and a functorial lift from SO $$(2n+1)$$ . The latter condition can be equivalently reformulated that the exterior square L-function of $$\Pi $$ has a pole at $$s=1$$ . In this paper, we prove a rationality result for the residue of the exterior square L-function at $$s=1$$ and also for the holomorphic value of the symmetric square L-function at $$s=1$$ attached to $$\Pi $$ . As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gaus sums of Hecke characters.

Let F be a totally real number field and let π be a cuspidal automorphic representation of GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} , which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period p ⁢ ( π ) {p(\pi)} to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods p ⁢ ( π ) {p(\pi)} and all of their twists p ⁢ ( π ⊗ ξ ) {p(\pi\otimes\xi)} under arbitrary algebraic Hecke characters ξ. In the appendix, we show that ( 𝔤 , K ) {(\mathfrak{g},K)} -cohomological cusp forms of GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} all qualify to be of the above type – providing a large source of examples. We expect that these period relations for GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, which we hope to report on in Part II of this paper.

Let $F$ be a totally real number field and $E/F$ a totally imaginary quadratic extension of $F$. Let $\Pi$ be a cohomological, conjugate self-dual cuspidal automorphic representation of $GL_n(\mathbb A_E)$. 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 $F = \mathbb Q$., Comment: This version of the paper has been entirely reworked. Its results are a "basis-free" variant of the previous version

Let${\mathcal{K}}$be an imaginary quadratic field. Let${\rm\Pi}$and${\rm\Pi}^{\prime }$be irreducible generic cohomological automorphic representation of$\text{GL}(n)/{\mathcal{K}}$and$\text{GL}(n-1)/{\mathcal{K}}$, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called aWhittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if${\rm\Pi}$is cuspidal and the weights of${\rm\Pi}$and${\rm\Pi}^{\prime }$are in a standard relative position, the critical values of the Rankin–Selberg product$L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$are essentially algebraic multiples of the product of the Whittaker periods of${\rm\Pi}$and${\rm\Pi}^{\prime }$. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal${\rm\Pi}$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.

We prove an arithmeticity result for a class of cohomological residual automorphic representations of a general reductive group G. More precisely, we show that this class of residual representations is stable under the action of Aut(C). This complements numerous results on the stability of cohomological cuspidal automorphic representations due by several people. We conclude by showing that the rationality eld of such a cohomological residual automorphic representation is a number eld.

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 $\Pi$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\Bbb{A})$ 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 behavior 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 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 ${\rm GL}_3\times{\rm GL}_2$; assuming Langlands Functoriality, this generalizes to certain Rankin-Selberg $L$-functions of ${\rm GL}_n\times{\rm GL}_{n-1}$. 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.

Let $G$ be a connected, reductive algebraic group over a number field $F$ and let $E$ be an algebraic representation of ${G}_{\infty } $. In this paper we describe the Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ of $G$ below a certain degree ${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map ${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$, $q\lt {q}_{ \mathsf{res} } $, for all automorphic representations $\Pi $ of $G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree ${q}_{ \mathsf{max} } $, the space of Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ 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 ${\mathrm{GL} }_{n} $ and the split classical groups of type ${B}_{n} $, ${C}_{n} $, ${D}_{n} $.

Let G/ℚ be an inner form of Sp4/ℚ which does not split over ℝ. Consequently, it is not quasi-split. In this paper we determine completely the automorphic cohomology of G. That is, we describe the Eisenstein and the cuspidal cohomology of congruence subgroups Γ of G with respect to arbitrary coefficient systems. In particular we establish precise nonvanishing results for cuspidal cohomology. In addition, we calculate the residual spectrum [Formula: see text] of G.

LetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

Let \({G/\mathbb Q}\) be the simple algebraic group Sp(n, 1) and \({\Gamma=\Gamma(N)}\) a principal congruence subgroup of level N ≥ 3. Denote by K a maximal compact subgroup of the real Lie group \({G(\mathbb R)}\) . Then a double quotient \({\Gamma\backslash G(\mathbb R)/K}\) is called an arithmetically defined, quaternionic hyperbolic n-manifold. In this paper we give an explicit growth condition for the dimension of cuspidal cohomology \({H^{2n}_{cusp}(\Gamma\backslash G(\mathbb R)/K,E)}\) in terms of the underlying arithmetic structure of G and certain values of zeta-functions. These results rely on the work of Arakawa (Automorphic Forms of Several Variables: Taniguchi Symposium, Katata, 1983, eds. I. Satake and Y. Morita (Birkhauser, Boston), pp. 1–48, 1984).

Let G = Sp4=k be the k-split symplectic group of k-rank 2, where k is a totally real numbereld. In this paper we compute the Eisenstein cohomology of G with respect to anynite{ dimensional, irreducible, k-rational representation E of G1 = Rk=QG( R), where Rk=Q denotes the restriction of scalars from k to Q. The approach is based on the work of Schwermer regarding the Eisenstein cohomology for Sp4=Q, Kim's description of the residual spectrum of Sp4, and the Frankeltration of the space of automorphic forms. In fact, taking the representation theoretic point of view, we write, for the group G, the Frankeltration with respect to the cuspidal support, and give a precise description of theltration quotients in terms of induced representations. This is then used as a prerequisite for the explicit computation of the Eisenstein cohomology. The special focus is on the residual Eisenstein cohomology. Under a certain compatibility condition for the coefficient system E and the cuspidal support, we prove the existence of non{trivial residual Eisenstein cohomology classes, which are not square{integrabl e, that is, represented by a non{ square{integrabl residue of an Eisenstein series.

The paper investigates a significant part of the automorphic, in fact of the so-called Eisenstein cohomology of split odd orthogonal groups over Q. The main result provides a description of residual and regular Eisenstein cohomology classes for maximal parabolic Q-subgroups in case of generic cohomological cuspidal automorphic representations of their Levi subgroups. That is, such identifying necessary conditions on these latter representations as well as on the complex parameters in order for the associated Eisenstein series to possibly yield non-trivial classes in the automorphic cohomology., 21 pages, some minor corrections made, journal reference added