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Elements of the Representation Theory of the Jacobi Group (Paperback, 2nd Revised edition)
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Elements of the Representation Theory of the Jacobi Group (Paperback, 2nd Revised edition)
Series: Modern Birkhauser Classics
Expected to ship within 10 - 15 working days
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After Pyatetski-Shapiro[PS1] and Satake [Sa1] introduced,
independent of one another, an early form of the Jacobi Theory in
1969 (while not naming it as such), this theory was given a de?nite
push by the book The Theory of Jacobi Forms by Eichler and Zagier
in 1985. Now, there are some overview articles describing the
developments in the theory of the Jacobigroupandits autom- phic
forms, for instance by Skoruppa[Sk2], Berndt [Be5] and Kohnen [Ko].
We
refertotheseformorehistoricaldetailsandmanymorenamesofauthorsactive
inthistheory,whichstretchesnowfromnumbertheoryandalgebraicgeometry
to theoretical physics. But let us only brie?y indicate several-
sometimes very closely related - topics touched by Jacobi theory as
we see it: *
?eldsofmeromorphicandrationalfunctionsontheuniversalellipticcurve
resp. universal abelian variety * structure and projective
embeddings of certain algebraic varieties and homogeneous spaces *
correspondences between di?erent kinds of modular forms *
L-functions associated to di?erent kinds of modular forms and
autom- phic representations * induced representations * invariant
di?erential operators * structure of Hecke algebras * determination
of generalized Kac-Moody algebras and as a ? nal goal related to
the here ?rst mentioned * mixed Shimura varieties and mixed
motives. Now, letting completely aside the arithmetical and
algebraic geometrical - proach to Jacobi forms developed and
instrumentalized by Kramer [Kr], we ix x Introduction will treat
here a certain representation theoretic point of view for the
Jacobi theory parallel to the theory of Jacquet-Langlands [JL] for
GL(2) as reported by Godement [Go2], Gelbart [Ge1] and, recently,
Bump [Bu].
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