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A Converse Theorem for Jacobi Forms

โœ Scribed by Yves Martin

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
634 KB
Volume
61
Category
Article
ISSN
0022-314X

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โœฆ Synopsis

Let f (q { , q z )= n, r c(n, r) q n { q r z be a power series whose coefficients satisfy a particular periodicity condition depending on the integer r modulo 2m. We first associate to f (q { , q z ) a 2m-vector-valued function 4( f, s) via a generalized Mellin transform. Then we show that the function 4( f, s) is entire, bounded on vertical strips and satisfies certain matrix functional equation if, and only if, f (q { , q z ) is the Fourier expansion of a Jacobi cusp form of index m invariant under the group SL(2, Z) _ Z 2 . This is the direct analogue of Hecke's converse theorem for elliptic cusp forms in the context of Jacobi cusp forms on SL(2, Z) _ Z 2 . 1996 Academic Press, Inc.

c r0 (N) N s , 0 r 0 2m&1, (2) article no. 0143

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