Geodesic
On definition fields of an algebraic curve
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 219-230 
A. L. Smirnov. On definition fields of an algebraic curve. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 219-230. http://geodesic.mathdoc.fr/item/ZNSL_2014_430_a13/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is considered such geometric invariants of an algebraic curve as the minimal number of crucial values of the rational functions and the minimal transcendence degree of the definition fields. The question is if the difference of these two invariants is always equal to 3 for any curve with the genus $g>0$. For curves defined over an algebraic number field the positive answer is given by Belyi's theorem. In the paper the positive answer is given for some other cases.

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