How big is the image of the Galois representations attached to CM elliptic curves?
Francesco Campagna, Riccardo Pengo

TL;DR
This paper provides a method to compute the size of the Galois image for CM elliptic curves over number fields, extending Serre's open image theorem through classical complex multiplication theory.
Contribution
It introduces a closed formula for calculating the Galois image size of CM elliptic curves, linking classical complex multiplication theory with modern Galois representation analysis.
Findings
Derived a closed formula for (E/F)
Connected classical CM theory with Galois image size computation
Enabled explicit calculation of Galois representation images for CM elliptic curves
Abstract
Using an analogue of Serre's open image theorem for elliptic curves with complex multiplication, one can associate to each CM elliptic curve defined over a number field a natural number which describes how big the image of the Galois representation associated to is. We show how one can compute , using a closed formula that we obtain from the classical theory of complex multiplication.
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Taxonomy
TopicsCryptography and Residue Arithmetic · Algebraic Geometry and Number Theory · Polynomial and algebraic computation
