Extensions of CM elliptic curves and orbit counting on the projective line

TL;DR

This paper connects formulas for orbit counts on the projective line to elliptic curve geometry, providing a general formula involving isogeny volcanoes and describing extensions of CM elliptic curves.

Contribution

It offers a new geometric interpretation of orbit formulas and introduces a comprehensive formula for congruence subgroups of Bianchi groups using isogeny volcanoes.

Findings

Interpretation of orbit formulas via elliptic curve geometry

A general formula involving walks on isogeny volcano graphs

Complete description of extensions of CM elliptic curves

Abstract

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, and of a pair of lattices in a quadratic field.