Harder's conjecture I

Abstract

Let 𝑓 be a primitive form with respect to SL₂(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 Sp₂(ℤ). This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.