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Zapiski Nauchnykh Seminarov POMI, 2014, Volume 430, Pages 219–230 (Mi znsl6091)  
On definition fields of an algebraic curve

A. L. Smirnov

St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia
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Abstract: 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.
Key words and phrases: algebraic curve, definition field, Belyi theorem, projective line, transcendence degree, ramification, critical value, rational function.
Received: 30.09.2014
English version:
Journal of Mathematical Sciences (New York), 2016, Volume 219, Issue 3, Pages 484–491
DOI: https://doi.org/10.1007/s10958-016-3121-6
Bibliographic databases:
Document Type: Article
UDC: 512.75
Language: Russian
Citation: A. L. Smirnov, “On definition fields of an algebraic curve”, Problems in the theory of representations of algebras and groups. Part 27, Zap. Nauchn. Sem. POMI, 430, POMI, St. Petersburg, 2014, 219–230; J. Math. Sci. (N. Y.), 219:3 (2016), 484–491
Citation in format AMSBIB
\Bibitem{Smi14}
\by A.~L.~Smirnov
\paper On definition fields of an algebraic curve
\inbook Problems in the theory of representations of algebras and groups. Part~27
\serial Zap. Nauchn. Sem. POMI
\yr 2014
\vol 430
\pages 219--230
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6091}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3486770}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 219
\issue 3
\pages 484--491
\crossref{https://doi.org/10.1007/s10958-016-3121-6}
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