  We examine crossing probabilities and free energies for conformally invariant
critical 2-D systems in rectangular geometries, derived via conformal field
theory and Stochastic L\"owner Evolution methods. These quantities are shown to
exhibit interesting modular behavior, although the physical meaning of modular
transformations in this context is not clear. We show that in many cases these
functions are completely characterized by very simple transformation
properties. In particular, Cardy's function for the percolation crossing
probability (including the conformal dimension 1/3), follows from a simple
modular argument. A new type of "higher-order modular form" arises and its
properties are discussed briefly.
