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...