A local criterion for the Saito-Kurokawa lifting of cuspforms with characters

Let $f$ be a holomorphic degree-2 Siegel cuspform of weight $\kappa$, level $N$, and nebentype a primitive Dirichlet character $\chi$. Let $f$ be an eigenfunction of the regular Hecke operators $T_p,T_{p^2}$ at primes $p\nmid N$ and an eigenfunction of the Frobenius operators $\Pi_p$ and their duals $\Pi^*_p$ at primes $p|N$. For certain $\chi$, we give conditions on the Satake parameters of $f$ which imply that $f$ is lifted from an elliptic cuspform $\phi$ of weight $2\kappa-2$, level $N$, and nebentype $\chi^2$. We also show that for such $f$ and $\phi$, the eigenvalues of the Frobenius operators on $f$ are eigenvalues of Hecke operators on $\phi$.

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Published 1 January 2009