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arXiv · 1608.01390

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

Julian Rosen·Ariel Shnidman
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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, and of a pair of lattices in a quadratic field.

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Julian Rosen, Ariel Shnidman. 2016-08-03. Extensions of CM elliptic curves and orbit counting on the projective line. https://doi.org/10.1007/s40993-017-0073-y

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