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The theory of Jacobi forms

Author: M Eichler; Don Zagier
Publisher: Boston : Birkhäuser, 1985.
Series: Progress in mathematics (Boston, Mass.), v. 55.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
The functions studied in this monogra9h are a cross between elliptic functions and modular forms in one variable. Specifically, we define a Jacobi form on SL (~) to be a holomorphic function 2 (JC = upper half-plane) satisfying the t\-10 transformation eouations 2Tiimcz· k CT +d a-r +b z) (1) ((cT+d) e cp(T, z) cp CT +d ' CT +d (2) rjl(T, z+h+]l) and having a Four·ier expansion of the form 00 e2Tii(nT +rz) (3) cp(T,  Read more...
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Details

Genre/Form: Electronic books
Additional Physical Format: Print version:
Eichler, M. (Martin).
Theory of Jacobi forms.
Boston : Birkhäuser, 1985
(DLC) 84028250
(OCoLC)11518281
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: M Eichler; Don Zagier
ISBN: 9781468491623 1468491628 9781468491647 1468491644
OCLC Number: 679034587
Reproduction Notes: Electronic reproduction. [S.l.] : HathiTrust Digital Library, 2010. MiAaHDL
Description: 1 online resource (v, 148 pages).
Details: Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002.
Series Title: Progress in mathematics (Boston, Mass.), v. 55.
Responsibility: Martin Eichler, Don Zagier.

Abstract:

The functions studied in this monogra9h are a cross between elliptic functions and modular forms in one variable. Specifically, we define a Jacobi form on SL (~) to be a holomorphic function 2 (JC = upper half-plane) satisfying the t\-10 transformation eouations 2Tiimcz· k CT +d a-r +b z) (1) ((cT+d) e cp(T, z) cp CT +d ' CT +d (2) rjl(T, z+h+]l) and having a Four·ier expansion of the form 00 e2Tii(nT +rz) (3) cp(T, z) 2: c(n, r) 2:: rE~ n=O 2 r ~ 4nm Here k and m are natural numbers, called the weight and index of rp, respectively. Note that th e function cp (T, 0) is an ordinary modular formofweight k, whileforfixed T thefunction z-+rjl( -r, z) isa function of the type normally used to embed the elliptic curve ~/~T + ~ into a projective space. If m= 0, then cp is independent of z and the definition reduces to the usual notion of modular forms in one variable. We give three other examples of situations where functions satisfying (1)-(3) arise classically: 1. Theta series. Let Q: ~-+ ~ be a positive definite integer valued quadratic form and B the associated bilinear form.

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