The quantum Hall curve

We show how the modular symmetries that have been found to be consistent with most available scaling data from quantum Hall systems, derive from a rigid family of algebraic curves of the elliptic type. The complicated special functions needed to describe scaling data arise in a simple and transparent way from the group theory and geometry of these \emph{quantum Hall curves}. The renormalization-group potential therefore emerges naturally in a geometric context that complements the phenomenology found in our companion paper [Phys. Rev. B 85, 155123 (2012)]. We show how the algebraic geometry of elliptic curves is an efficient way to analyze specific scaling data, extract the modular symmetries of the transport coefficients, and use this information to fit the given system into the one-dimensional (real) family of curves that may model all universal properties of quantum Hall systems.