Canonical models of modular curves and the Galois action on CM-points

TL;DR

This paper explores the canonical models of modular curves and characterizes the Galois group of the field generated by all CM-points, revealing its structure as an extension of an abelian group by a 2-element group.

Contribution

It provides a detailed description of the projective limit of modular curves and the Galois action on CM-points using Shimura variety theory.

Findings

Galois group of Q(CM) over Q is an extension of an abelian group by a 2-element group

Describes the automorphism group of the projective limit of Y(N) curves

Connects canonical models of Shimura varieties with Galois actions on CM-points

Abstract

We use the theory of canonical models of Shimura varieties to describe the projective limit of the curves Y(N), all N, and its automorphism group. In particular we prove that the Galois group of Q(CM) over Q is an extension of a certain abelian group by a 2-element group, where Q(CM) stands for the the extension of Q by all the CM-points on all the curves Y(N).