On the cuspidality of pullbacks of Siegel Eisenstein series to Sp2m×Sp2n

In this paper we study the pullback of a Siegel Eisenstein series on Sp2m+2n to Sp2m×Sp2n. There is a well-established literature on such pullbacks. In the case that m=n Garrett showed that the pullback is actually a cusp form in each variable separately. Here we generalize this result showing the pullback is cuspidal in the smaller variable in the case m≠n. Such results have applications to producing congruences between Siegel modular forms.