  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.
