Harder's conjecture I

TL;DR

This paper proposes a conjecture relating certain lifts of modular forms, which implies Harder's conjecture, and proves it in specific cases, advancing understanding in the theory of automorphic forms.

Contribution

It introduces a new conjecture connecting Klingen-Eisenstein lifts and vector valued Hecke eigenforms, providing partial proofs and implications for Harder's conjecture.

Findings

Conjecture implies Harder's conjecture.

Proved the conjecture in some cases.

Establishes links between different types of lifts of modular forms.

Abstract

Let $f$ be a primitive form with respect to $SL_2(Z)$. 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 $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.