On the minimal field of definition of rational maps: Rational maps of odd signature

Abstract

The field of moduli of a rational map is an invariant under conjugation by Mobius 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.