{"product_id":"the-theory-of-jacobi-forms-von-martin-eichler-don-zagier","title":"The Theory of Jacobi Forms","description":"\u003cp\u003eThe 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.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9781468491647\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9781468491647","offer_id":39416975294557,"sku":"9781468491647","price":128.39,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/1522d879-46af-4c0d-b2db-cc4722871845.jpg?v=1772169081","url":"https:\/\/shop.autorenwelt.de\/products\/the-theory-of-jacobi-forms-von-martin-eichler-don-zagier","provider":"Autorenwelt Shop","version":"1.0","type":"link"}