{"product_id":"the-theory-of-jacobi-forms-martin-eichler-9781468491647","title":"The Theory of Jacobi Forms","description":"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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e Martin Eichler, Don Zagier\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 1468491644\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9781468491647\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Birkhauser\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 09\/14\/2013\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 150\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Paperback\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 0.64lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 10.00h x 7.00w x 0.34d","brand":"Martin Eichler","offers":[{"title":"Paperback","offer_id":43935438078207,"sku":"9781468491647","price":129.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/products\/img_0345375b-7959-43f1-aa05-c23a99289b97.jpg?v=1681419797","url":"https:\/\/www.whiterainbookhouse.com\/products\/the-theory-of-jacobi-forms-martin-eichler-9781468491647","provider":"WR Book House","version":"1.0","type":"link"}