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

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