Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a totally real field?
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9$\begingroup$ This is a theorem of L. Moret-Bailly. See "Groupes de Picard et problemes de Skolem I, II", Ann. Sci. ENS Ser. 4, 1989. It does not require your geometric restrictions on $X$. $\endgroup$– Vesselin DimitrovCommented Mar 11, 2014 at 18:18
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1$\begingroup$ For an exposition of the proof I can also refer you to Brian Conrad's "Overview of Moret-Bailly's theorem on global points." $\endgroup$– Vesselin DimitrovCommented Mar 11, 2014 at 18:39
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2$\begingroup$ The document to which Dimitrov refers is at this link: math.stanford.edu/~conrad/vigregroup/vigre05/mb.pdf $\endgroup$– user76758Commented Mar 12, 2014 at 1:14
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