Weights of the Mod p Kernels of Theta Operators

Abstract

AbstractLet Θ[j] be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime p, we give the weights of elements of mod p kernel of Θ[j], where the mod p kernel of Θ[j] is the set of all Siegel modular forms F such that Θ[j](F) is congruent to zero modulo p. In order to construct examples of the mod p kernel of Θ[j] fromany Siegel modular form, we introduce new operators A(j)(M) and show the modularity of F|A(j)(M) when F is a Siegel modular form. Finally, we give some examples of the mod p kernel of Θ[j] and the filtrations of some of them.