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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.
| Autorius: | Don Zagier, Martin Eichler, |
| Serija: | Progress in Mathematics |
| Leidėjas: | Birkhäuser Boston |
| Išleidimo metai: | 2013 |
| Knygos puslapių skaičius: | 160 |
| ISBN-10: | 1468491644 |
| ISBN-13: | 9781468491647 |
| Formatas: | 254 x 178 x 9 mm. Knyga minkštu viršeliu |
| Kalba: | Anglų |
Parašykite atsiliepimą apie „The Theory of Jacobi Forms“
