Let ${{\Theta }^{[j]}}$ 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 ${{\Theta }^{[j]}}$ , where the mod $p$ kernel of ${{\Theta }^{[j]}}$ is the set of all Siegel modular forms $F$ such that ${{\Theta }^{[j]}}(F)$ is congruent to zero modulo $p$ . In order to construct examples of the mod $p$ kernel of ${{\Theta }^{[j]}}$ from any Siegel modular forms, we introduce new operators ${{A}^{(j)}}(M)$ and show the modularity of $F|{{A}^{\left( j \right)}}\left( M \right)$ when $F$ is a Siegel modular form. Finally, we give some examples of the mod $p$ kernel of ${{\Theta }^{[j]}}$ and the filtrations of some of them.

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$ .

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.