On Siegel Maass Wave Forms of Weight 0
thesisposted on 28.06.2013, 00:00 by Christine A. Robinson
Progress has been made toward a Saito-Kurokawa lift, including a non-holomorphic Shimura lift and a lift from the non-holomorphic analogue of the Kohnen plus space to Jacobi Maass forms. A large gap remains in our understanding of Siegel Maass forms, which are the non-holomorphic analogue of Siegel modular forms. Relatively few results are known with a high degree of generality, and even basic results have not been developed in some cases. In the case of Siegel Maass wave forms of weight 0, Niwa, in 1991, utilized explicit differential operators given by Nakajima (1982) to develop the Fourier series expansion. However, Nakajima's quartic differential operator is not invariant under the action of the desired slash operator, and so we still lack a valid Fourier expansion for Siegel Maass wave forms of weight 0. In this thesis, we introduce Siegel Maass wave forms of weight 0, which are simultaneous eigenvectors of Maass' Casimir operators, rather than the operators given by Nakajima, and follow the method of Niwa to obtain a fourth order ordinary differential equation, which must be satisfied by the Fourier coefficients of such wave forms. In Chapter 2, we review the theory of holomorphic Siegel modular forms and the classical Saito-Kurokawa lift. In Section 3.1, we define Siegel Maass wave forms of weight 0, and in Section 3.2, we describe non-holomorphic automorphic forms involved in a Saito-Kurokawa lift, as well as the maps between them which have previously been established. In Section 4.1, we explicitly compute the Casimir operators which form the basis for our definition of Siegel Maass wave forms, followed by the computation of the system of differential equations satisfied by these forms, in Section 4.2. In Section 4.3, through a series of changes of variable, we reduce this system of differential equations to a single fourth order ordinary linear differential equation. The Fourier coefficient of a wave form, corresponding to the identity matrix, will satisfy this differential equation. We discuss the proof given by Niwa for the solutions to his ordinary differential equation and his method for obtaining the first solution by theta lifting, in Section 4.4, and finally we conclude in Section 4.5 by giving the Fourier coefficients corresponding to definite matrices, according to Hori.