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Let $X$ be an anabelian curve over a number field $K$ and let $p:Y\rightarrow X$ be a finite etale cover. Then is anything known (or has anything been conjectured) about the field of definition of $Y$? In particular, should it be an abelian extension of $K$?

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see the answer to my question,… which implies that you can get really awful extensions this way. – Will Sawin Feb 25 '13 at 17:06
up vote 3 down vote accepted

Certainly not, unless I totally misunderstand your question. For instance, an unramified double cover of X is going to be defined over the field obtained by adjoining a 2-torsion point of the Jacobian of X, and that field is going to be non-abelian as all get out.

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