What question did this study set out to answer?

The aim is to establish new bijections between different types of Siegel modular forms and reinterpret congruences for half-integral weights.

April 4, 2026Open Access

Conjectures of Shimura Type and of Harder Type Revisited

Key Points

The aim is to establish new bijections between different types of Siegel modular forms and reinterpret congruences for half-integral weights.
Propose conjectures linking half-integral and integral weight Siegel modular forms.
Generalize previous conjectures related to Neben type forms.
Provide concrete examples demonstrating the validity of conjectures.
Establish bijective correspondence between spaces of Siegel modular forms of half-integral and integral weights.
Verify the Harder conjecture's interpretation for half-integral weight forms with concrete examples.

Abstract

We propose a conjecture of Shimura type that the level one part of the space of vector valued Siegel modular forms of degree two of half integral weight without character (Haupt type) corresponds bijectively, up to liftings, to the space of vector valued Siegel modular forms of integral weight of degree two of level one.This is a generalization of our previous conjecture for Neben type (with character).Together with the previous conjecture, this means that Siegel modular forms of degree two of half integral weight with character and without character should correspond bijectively and Hecke equivariantly up to liftings.The Harder conjecture on congruences for vector valued Siegel cusp forms of integral weight is now interpreted as a half-integral weight version which means the congruence between eigenvalues of Siegel cusp forms and non-cusp forms of half-integral weight of the same group.We give a concrete example that this congruence really holds.

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