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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asked Feb 25, 2013 at 15:49
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2$\begingroup$ see the answer to my question, mathoverflow.net/questions/119981/… which implies that you can get really awful extensions this way. $\endgroup$– Will SawinCommented Feb 25, 2013 at 17:06
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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.
