InfoMod: a visual and computational approach to Gauss’ binary quadratic forms

InfoMod is a software devoted to the modular group, PSL2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {PSL}_2 (\mathbf {Z})$$\end{document}. It consists of algorithms that deal with the classical correspondences among geodesics on the modular surface, elements of the modular group and binary quadratic forms. In addition, the software implements the recently discovered representation of Gauss’ indefinite binary quadratic forms and their classes in terms of certain infinite planar graphs (dessins) called çarks. InfoMod illustrates various aspects of these forms, i.e. Gauss’ reduction algorithm, the representation problem of forms, ambiguous and reciprocal forms. It can be used for visualization, for high performance computation involving these mathematical structures as well as experimenting.