Weights of the mod $p$ kernel of the theta operators

Abstract

We give some relations between the weights and the prime $p$ of elements of the mod $p$ kernel of the generalized theta operator ${\Theta }^{[j]}$. In order to construct examples of the mod $p$ kernel of ${\Theta }^{[j]}$ from any modular form, we introduce new operators $A^{(j)}(M)$ and show the modularity of $F|A^{(j)}(M)$ when $F$ is a modular form. Finally, we give some examples of the mod $p$ kernel of ${\Theta }^{[j]}$ and the filtrations of some of them.

28 pages, 3 tables