Pullback of the lifting of elliptic cusp forms and Miyawaki's conjecture

Abstract

We construct a lifting from Siegel cusp forms of degree $r$ to Siegel cusp forms of degree $r+2n$. For $r=n=1$, our result is a partial solution of a conjecture made by Miyawaki [27, page 307] in 1992. In particular, we can calculate the standard $L$-function of a cusp form of degree 3 and weight 12, which is in accordance with Miyawaki's conjecture. We give a conjecture on the Petersson inner product of the lifting in terms of certain $L$-values