Periods of automorphic forms: The case of (U n+1×U n ,U n )

2019 ◽  
Vol 2019 (746) ◽  
pp. 1-38 ◽  
Author(s):  
Atsushi Ichino ◽  
Shunsuke Yamana
Keyword(s):  
Eisenstein Series ◽  
Automorphic Forms ◽  
Diagonal Subgroup ◽  
Periods Of Automorphic Forms

Abstract Following Jacquet, Lapid and Rogawski, we define regularized periods of automorphic forms on \mathrm{U}_{n+1} \times \mathrm{U}_{n} along the diagonal subgroup {\mathrm{U}_{n}} and compute the regularized periods of cuspidal Eisenstein series and their residues. The formula for the periods of residues has an application to the Gan–Gross–Prasad conjecture.

scholarly journals Periods of automorphic forms: the case of

2014 ◽  
Vol 151 (4) ◽  
pp. 665-712 ◽  
Author(s):  
Atsushi Ichino ◽  
Shunsuke Yamana
Keyword(s):  
Automorphic Form ◽  
Automorphic Forms ◽  
Selberg Integral ◽  
Selberg Integrals ◽  
Diagonal Subgroup ◽  
Periods Of Automorphic Forms

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on $\text{GL}_{n+1}\times \text{GL}_{n}$ along the diagonal subgroup $\text{GL}_{n}$ and express it in terms of the Rankin–Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of Rankin–Selberg integrals to all automorphic forms on $\text{GL}_{n+1}\times \text{GL}_{n}$.

PERIODS OF AUTOMORPHIC FORMS: THE TRILINEAR CASE

2015 ◽  
Vol 17 (1) ◽  
pp. 59-74 ◽  
Author(s):  
Shunsuke Yamana
Keyword(s):  
Eisenstein Series ◽  
Automorphic Forms ◽  
Metaplectic Groups ◽  
Periods Of Automorphic Forms

Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe 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.

scholarly journals Residues of Eisenstein series and the automorphic cohomology of reductive groups

2013 ◽  
Vol 149 (7) ◽  
pp. 1061-1090 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Number Field ◽  
Automorphic Forms ◽  
Reductive Groups ◽  
Algebraic Representation ◽  
Reductive Algebraic Group ◽  
Automorphic Representations ◽  
Residual Spectrum ◽  
Eisenstein Cohomology

AbstractLet $G$ 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} $.

Sign in / Sign up

Export Citation Format

Share Document