By Martin Eichler, Don Zagier

The services studied during this monogra9h are a pass among elliptic services and modular kinds in a single variable. in particular, we outline a Jacobi shape on SL ( ) to be a holomorphic functionality 2 (JC = top half-plane) gratifying the t\-10 transformation eouations 2Tiimcz. ok 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 enlargement of the shape 00 e2Tii(nT +rz) (3) cp(T, z) 2: c(n, r) 2:: rE n=O 2 r 4nm the following ok and m are typical numbers, referred to as the load and index of rp, respectively. observe that th e functionality cp (T, zero) is a typical modular formofweight okay, whileforfixed T thefunction z-+rjl(-r, z) isa functionality of the kind as a rule used to embed the elliptic curve / T + right into a projective house. If m= zero, then cp is self sustaining of z and the definition reduces to the standard proposal of modular varieties in a single variable. We supply 3 different examples of events the place features pleasurable (1)-(3) come up classically: 1. Theta sequence. permit Q: -+ be a favorable yes integer valued quadratic shape and B the linked bilinear shape.