Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 685-692

ON THE MINIMAL FIELD OF DEFINITION OF RATIONAL MAPS: RATIONAL MAPS OF ODD SIGNATURE

Rubén A. Hidalgo

Universidad de La Frontera, Departamento de Matemática y Estadística
Temuco, Chile; ruben.hidalgo 'at' ufrontera.cl

Abstract. The field of moduli of a rational map is an invariant under conjugation by Möbius transformations. Silverman proved that a rational map, either of even degree or equivalent to a polynomial, is definable over its field of moduli and he also provided examples of rational maps of odd degree for which such a property fails. We introduce the notion for a rational map to have odd signature and prove that this condition ensures for the field of moduli to be a field of definition. Rational maps being either of even degree or equivalent to polynomials are examples of odd signature ones.

2010 Mathematics Subject Classification: Primary 37P05, 37F10, 14G05.

Key words: Rational maps, field of moduli, field of definition, Galois groups.

Reference to this article: R. A. Hidalgo: On the minimal field of definition of rational maps: rational maps of odd signature. Ann. Acad. Sci. Fenn. Math. 43 (2018), 685-692.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4340