scholarly journals On generalized theta series liftings

1997 ◽  
Vol 62 (2) ◽  
pp. 229-238
Author(s):  
Min Ho Lee
Keyword(s):  
Automorphic Forms ◽  
Abelian Varieties ◽  
Theta Series ◽  
Symplectic Groups ◽  
Reductive Groups ◽  
Generalized Theta Series

AbstractWe generalize dual reductive pairs by using reductive groups that are not necessarily subgroups of symplectic groups and construct the corresponding theta-series liftings for certain types of automorphic forms. We also discuss connections of such generalized theta-series liftings with families of abelian varieties parametrized by an arithmetic variety.

scholarly journals Weil type representations and automorphic forms

1980 ◽  
Vol 77 ◽  
pp. 145-166 ◽  
Author(s):  
Toshiaki Suzuki
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Theta Series ◽  
Type Representation ◽  
Generalized Poisson ◽  
Summation Formulae ◽  
Reciprocity Law ◽  
Weil Type ◽  
Generalized Theta Series ◽  
Poisson Summation

During 1934-1936, W. L. Ferrar investigated the relation between summation formulae and Dirichlet series with functional equations, inspired by Voronoi’s works (1904) on summation formula with respect to the numbers of divisors. In [11] A. Weil showed that the automorphic properties of theta series are expressed by generalized Poisson summation formulae with respect to the so-called Weil representation. On the other hand, T. Kubota, in his study of the reciprocity law in a number field, defined a generalized theta series and a generalized Weil type representation of SL(2, C) and obtained similar results to those of A. Weil (1970-1976, [5], [6], [7]). The methods, used by W. L. Ferrar and T. Kubota, to obtain (generalized Poisson) summation formulae depend similarly on functional equations of Dirichlet series (attached to the generalized theta series).

scholarly journals Siegel modular forms and theta series attached to quaternion algebras

1991 ◽  
Vol 121 ◽  
pp. 35-96 ◽  
Author(s):  
Siegfried Böcherer ◽  
Rainer Schulze-Pillot
Keyword(s):  
Modular Forms ◽  
Orthogonal Group ◽  
Automorphic Forms ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Theta Correspondence ◽  
Quaternion Algebras ◽  
Metaplectic Group

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras II

1997 ◽  
Vol 147 ◽  
pp. 71-106 ◽  
Author(s):  
S. Böcherer ◽  
R. Schulze-Pillot
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Automorphic Forms ◽  
Multiplicative Group ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Quaternion Algebras

AbstractWe continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

scholarly journals Poisson transforms and mixed automorphic forms on semisimple Lie groups

2000 ◽  
Vol 61 (3) ◽  
pp. 353-370
Author(s):  
Min Ho Lee ◽  
Hyo Chul Myung
Keyword(s):  
Lie Groups ◽  
Vector Bundles ◽  
Automorphic Forms ◽  
Abelian Varieties ◽  
Semisimple Lie Groups

We discuss Poisson transforms which carry sections of certain vector bundles to mixed automorphic forms, and identify vector bundles whose sections are liftings of holomorphic forms on families of Abelian varieties via Poisson transforms.

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} $.

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