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

Abstract

There are several formulas for the number of orbits of the projective line under the action of subgroups of $G{L}_2$. 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.

15 pages, 3 figures

References in corpus (2)

• Isogeny volcanoes
• Néron-Severi groups of product abelian surfaces

Cited by in corpus (3)

• Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families
• Eisenstein cocycles in motivic cohomology
• Quadratic fields, Artin-Schreier extensions, and Bell numbers