There are several formulas for the number of orbits of the projective line under the action of subgroups of GL 2. We give an interpretation of two such formulas in terms of the geometry of elliptic curves, and prove a more general formula for a large class of congruence subgroups of Bianchi groups. Our formula involves the number of walks on a certain graph called an isogeny volcano. Underlying our results is a complete description of the group of extensions of a pair of CM elliptic curves, as well as the group of extensions of a pair of lattices in a quadratic field.
Bibliographical noteFunding Information:
AS thanks Andrew Snowden for a helpful conversation and Andrew Sutherland for help with the figures. AS was partially supported by NSF Grant DMS-0943832. Open Access
© 2017, The Author(s).
- Complex multiplication
- Elliptic curves
- Kleinian groups