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Elements of the Representation Theory of the Jacobi Group (Paperback, 1998 ed.)
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Elements of the Representation Theory of the Jacobi Group (Paperback, 1998 ed.)
Series: Progress in Mathematics, 163
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After Pyatetski-Shapiro [PSI] and Satake [Sal] 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
definite 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 Jacobi group and its automor-
phic forms, for instance by Skoruppa [Sk2], Berndt [Be5] and Kohnen
[Ko]. We refer to these for more historical details and many more
names of authors active in this theory, which stretches now from
number theory and algebraic geometry to theoretical physics. But
let us only briefly indicate several - sometimes very closely
related - topics touched by Jacobi theory as we see it: * fields of
meromorphic and rational functions on the universal elliptic curve
resp. universal abelian variety * structure and projective
embeddings of certain algebraic varieties and homogeneous spaces *
correspondences between different kinds of modular forms *
L-functions associated to different kinds of modular forms and
autom- phic representations * induced representations * invariant
differential operators * structure of Hecke algebras *
determination of generalized Kac-Moody algebras and as a final goal
related to the here first mentioned * mixed Shimura varieties and
mixed motives.
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