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

Julian Rosen, Ariel Shnidman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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, as well as the group of extensions of a pair of lattices in a quadratic field.

Original languageAmerican English
Article number9
JournalResearch in Number Theory
Volume3
Issue number1
DOIs
StatePublished - 1 Dec 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017, The Author(s).

Keywords

  • Complex multiplication
  • Elliptic curves
  • Kleinian groups

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