Crossing probabilities, their densities, and modular forms

 A crossing probability is the probability of finding, in a physical model,
a critical cluster that touches specified boundary arcs; its density
conditions on a point z being in a specified cluster.  We consider various
examples, for percolation and related models, on a rectangle.
Surprisingly, all known crossing formulas have modular properties, being
either modular forms, second-order modular forms or transforming like
Hermitian Jacobi modular functions.  This is unexpected because a rectangle
lacks toroidal symmetry; the origin of this modular behavior is a mystery