Newforms of half-integral weight: the minus space of S_{k+1/2}({\Gamma}_0(8M))

TL;DR

This paper explores the structure of the minus space of half-integral weight modular forms, establishing its relation to newforms of integral weight and level, through the computation of specific Hecke algebras and operators.

Contribution

It introduces a new characterization of the minus space of half-integral weight forms via Hecke operators and establishes its isomorphism with the space of newforms of integral weight.

Findings

The minus space is a common -1 eigenspace of certain Hecke operators.

The minus space is isomorphic to the space of newforms of weight 2k and level 4M.

Forms in the minus space satisfy a specific Fourier coefficient condition.

Abstract

We compute generators and relations for a certain $2$-adic Hecke algebra of level $8$ associated with the double cover of $\mathrm{SL}_2$ and a $2$-adic Hecke algebra of level $4$ associated with $\mathrm{PGL}_2$. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the $2$-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight $k+1/2$ and level $8M$ where $M$ is odd and square-free. Using these operators and our previous results on half-integral weight forms of level $4M$ we define a subspace of the space of half-integral weight forms as a common $-1$ eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace which we call the minus space is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight $2k$ and level $4M$. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.