Progress in Mathematics

The Theory of Jacobi Forms

Authors: Eichler, Martin, Zagier, Don

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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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Buy this book

eBook $69.99 net
( price for USA )
  • ISBN 978-1-4684-9162-3
  • digitally watermarked, no DRM
  • included format: PDF
  • eBooks can be used on all Reading Devices
Softcover $99.00 net
( price for USA )
  • ISBN 978-1-4684-9164-7
  • free shipping for individuals worldwide
  • usually dispatched within 3 to 5 business days

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Bibliographic Information

Bibliographic Information
Book Title
The Theory of Jacobi Forms
Series Title
Progress in Mathematics
Series Volume
55
Copyright
1985
Publisher
Birkhäuser Basel
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4684-9162-3
DOI
10.1007/978-1-4684-9162-3
Softcover ISBN
978-1-4684-9164-7
Series ISSN
0743-1643
Edition Number
1
Topics