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Contemporary Mathematics
Fields of definition of canonical curves
D. D. Long and A. W. Reid
Abstract. This note studies the question of which number fields can arise as
canonical components.
1. Introduction
Let k C be a field. A complex algebraic set V Cn
is defined over k if the
ideal of polynomials I(V ) vanishing on V is generated by a subset of k[x1, . . . , xn].
We say that a field k is the field of definition of V if V is defined over k, and if for
any other field K C with V defined over K, then k K (for the existence of
the field of definition, see [14], Chapter III). Note that the field of definition of an
algebraic variety depends on the embedding in a particular Cn
. By a curve we will
mean an irreducible algebraic curve unless otherwise stated.
Now let M be an orientable finite volume hyperbolic 3manifold with cusps, and
let X(M) (resp. Y (M)) denote the SL(2, C)character variety (resp. PSL(2, C)
character variety) associated to 1(M) (see for example [7] and [2] for definitions).
In [7] and [2] it is shown that X(M) and Y (M) are defined over Q. However, the
