What question did this study set out to answer?

The aim is to propose a conjecture relating the Klingen-Eisenstein lift of the Miyawaki lift of two primitive forms and establish its implications for Harder's conjecture.

April 1, 2026Open Access

Harder's Conjecture and Miyawaki Lift

Key Points

The aim is to propose a conjecture relating the Klingen-Eisenstein lift of the Miyawaki lift of two primitive forms and establish its implications for Harder's conjecture.
Defined parameters k, j, and n to satisfy specific modular arithmetic conditions.
Ensured forms f and g meet the necessary criteria for SL2(Z) and Sp2(Z).
Developed a conjectural framework for the congruence between lifts of specific forms.
Demonstrated that the proposed conjecture implies Harder's conjecture in certain cases.
Established congruences between the Klingen-Eisenstein lift and vector-valued Hecke eigenforms.

Abstract

Let k, j and n be positive integers such that k is odd , and both j and n are even, satisfying j ≡ n mod 4. Let f and g be primitive forms of weight 2k + j - 2 and k+j/2 - n/2 - 1, respectively, for SL2(Z). Then, we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Miyawaki lift of f and g of type II and a certain lift of a vector-valued Hecke eigenform of weight (k + j, k) for Sp2(Z). This conjecture implies Harder's conjecture. Through this formulation, we prove Harder's conjecture in some cases.

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