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" The Theory of Jacobi Forms "
by Martin Eichler, Don Zagier.
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BL
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| Record Number
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574517
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b403736
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| Main Entry
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Eichler, Martin.
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| Title & Author
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The Theory of Jacobi Forms\ by Martin Eichler, Don Zagier.
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| Publication Statement
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Boston, MA :: Birkhäuser Boston :: Imprint: Birkhäuser,, 1985.
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| Series Statement
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Progress in Mathematics,; 55
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| ISBN
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9781468491623
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: 9781468491647
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| Abstract
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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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Mathematics.
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Geometry, Algebraic.
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Group theory.
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Functional analysis.
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Number theory.
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| Added Entry
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Zagier, Don.
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SpringerLink (Online service)
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