Newforms of half-integral weight: the minus space counterpart

TL;DR

This paper introduces a new subspace of half-integral weight modular forms, establishing an isomorphism with newforms of integral weight via the Shimura-Niwa correspondence, expanding understanding of modular form spaces.

Contribution

It defines a novel minus space of half-integral weight modular forms and proves its Hecke isomorphism with the space of newforms of integral weight.

Findings

The minus space is isomorphic to the space of newforms of weight 2k and level 2M.

The minus space is a proper subspace of the orthogonal complement of the Kohnen plus space.

The isomorphism is compatible with Hecke operators.

Abstract

We define a subspace of the space of holomorphic modular forms of weight $k+1/2$ and level $4M$ where $M$ is odd and square-free. We show that this subspace is isomorphic under the Shimura-Niwa correspondence to the space of newforms of weight $2k$ and level $2M$ and that this is a Hecke isomorphism. The space we define is a proper subspace of the orthogonal complement of the Kohnen plus space if the Kohnen plus space is nonzero.