## The Theory of Jacobi FormsThe 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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### Contents

1 | |

Notations | 7 |

Relations with Other Types of Modular Forms | 57 |

Jacobi theta series and a theorem of Waldspurger | 81 |

The Ring of Jacobi Forms | 89 |

Tables | 141 |

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calculation coefficient of q Cohen compute conditions at infinity congruence subgroup Corollary to Theorem corresponding cT+d cu(N cusp form defined denotes dim k,m dim Mk dimension Dirichlet character divisors eigenvalues Eisenstein series equals equation Euler product explicitly fact finite follows form of weight forms of half-integral forms of index formula Fourier coefficients Fourier development Fourier expansion free module given gives half-integral weight Hecke algebra Hecke eigenforms Hecke operators Heegner points hence identity injective integer invariant isomorphism Jk,m Kohnen law of Jacobi lower bounds Lp(s lu mod 2m Maass Math matrices meromorphic modular group modulo multiple non-zero obtain odd weight polynomial prime proof of Theorem prove Theorem quadratic form quotient r(mod 2m relation replaced resp result ring Siegel modular forms Spezialschar square-free Table Theorem 5.6 theta series transformation law vanishes weak Jacobi form write zero