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

We give some relations between the weights and the prime $p$ of elements of the mod $p$ kernel of the generalized theta operator $Θ^{[j]}$. In order to construct examples of the mod $p$ kernel of $Θ^{[j]}$ 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 $Θ^{[j]}$ and the filtrations of some of them.

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