General construction of minus genus-subspace modular forms

Construct a general method for producing elements of the minus genus subspace $M_{3/2}^{G,-}(Gamma_0(4N))$ of half-integral-weight modular forms associated with a genus $G$ of binary quadratic forms, beyond the examples obtained from theta functions.

Background

The paper decomposes spaces of modular forms into genus subspaces determined by the genus characters of binary quadratic forms. In the worked example for N=3N=3, the authors construct a natural element of the plus genus subspace using the theta function associated with the principal quadratic form, and then exhibit an element of the corresponding minus genus subspace through a specific linear combination of basis elements computed with Magma.

The authors explicitly distinguish this example from a general construction: theta functions naturally yield elements of the plus subspaces, whereas no general procedure is known for constructing elements of the minus subspaces. The unresolved problem is therefore to develop such a construction for M3/2G,−(Γ0(4N))M_{3/2}^{G,-}(\Gamma_0(4N)) in general.

References

Note that the above construction demonstrates how elements of $ M_{\frac{3}{2}}{G,+}(\Gamma_0(4N))$ may be naturally constructed via theta functions, whereas a general construction of elements in $ M_{\frac{3}{2}}{G,-}(\Gamma_0(4N))$ is not known.

— Hurwitz class number relations and mock modular forms  (2608.14371 - Bringmann et al., 14 Aug 2026) in Example in Section 4, “Genus subspaces of $M_\kappa(\Gamma_0(4N))$ and $H_\kappa(\Gamma_0(4N))$”