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

Julian Rosen; Ariel Shnidman

doi:10.1007/s40993-017-0073-y

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 journal › Article › peer-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 language	American English
Article number	9
Journal	Research in Number Theory
Volume	3
Issue number	1
DOIs	https://doi.org/10.1007/s40993-017-0073-y
State	Published - 1 Dec 2017
Externally published	Yes

Bibliographical note

Funding 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

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

Keywords

Complex multiplication
Elliptic curves
Kleinian groups

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10.1007/s40993-017-0073-y

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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.",

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AU - Shnidman, Ariel

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N2 - 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.

AB - 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.

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KW - Kleinian groups

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JO - Research in Number Theory

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ER -

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

Abstract

Bibliographical note

Keywords

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