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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
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 , 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 3-manifold 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  and  for definitions).
In  and  it is shown that X(M) and Y (M) are defined over Q. However, the