Weil representations of Hilbert modular groups

Abstract

Abstract. We know that the well-known double cover Mp(2,Z) of SL(2,Z)acts on the group algebra C[M] of maps from M to the complex numbersC. Here M is the underlying group of a given finite quadratic Z-module(M,Q). We call the representation afforded by this action the ’Weil rep-resentation associated to (M,Q)’. It is remarkable to note that due to arecent result when we consider Weil representations of finite quadraticmodules over number fields the double cover Mp(2,O) (O is the ring ofintegers of the number field in question), which is used in the theory ofHilbert modular forms of half integral weight, does not play the samerole as in the case of the rational number field. We observe that thereare more double covers available to satisfy this action in the general case.It actually depends on the splitting of the ideal generated by 2 in thenumber field. However, when we restrict ourselves to finite quadraticmodules which are discriminant modules of lattices over O we see thatthe group Mp(2,O) acts on C[M]. For proving this we realize the Weilrepresentations in question by theta functions.