Harder's conjecture I

Abstract

Let $f$ be a primitive form with respect to $\mathrm{SL}_2(\mathbb{Z})$. Then, we propose a conjecture on the congruence between the Klingen–Eisenstein lift of the Duke–Imamoglu–Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $\mathrm{Sp}_2(\mathbb{Z})$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.