Congruences for Γ1(4)-modular forms of half-integral weight

The main result of this paper gives a complete description of the algebra of level 4, mod p modular forms of half-integral weight. We show that this is the algebra of regular functions of an affine curve over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbb F}_p$$\end{document}. The result parallels results of Swinnerton-Dyer (level 1), and Katz (general level) for integral weight forms.