Automorphic Forms on the Expanded Symmetric Domain of Type IV

We introduce a lifting from a given Jacobi form of index 1 to an automorphic form on an expanded domain of type F , introduced by Saito [20, 22]. The method is a generalization of Gritsenko [10, 11] for symmetric domain of type IV. We constract a lifting function satisfying a certain translation formula on the expanded domain. §0. Introduction §1. Expanded symmetric domain of type 1Y §2. Expanded Jacobi forms §3. Lifting function of Jacobi forms §4. Arithmetic lifting on the expanded domain §0. Introduction It is classically known as the Jacobi's inversion problem to find a description of the coordination of a Riemann surface in terms of integrals on the Riemann surface. In case of an elliptic curve, this was solved by the use of elliptic modular functions. In general for higher genus case, this was solved by the use of theta functions (cf. Siegel [23]). One may consider similar problem for the integrals on higher dimensional varieties. In this case, the main problem is to construct automorphic forms on the domain of periods. The case for family of polarized #3 surfaces was studied by Pyatetski-Shapiro [18], where the domain of periods is the classical symmetric domain of type W. Even in this case, we do not have a full description of the ring of automorphic forms. Communicated by K. Saito, April 17, 1998. Revised October 16, 1998 1991 Mathematics Subject Classifications: 11F55, 14J17 *Graduate School of Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502,Japan, e-mail: hiroki@kurims.kyoto-u.ac.jp