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Fields of moduli and fields of definition of odd signature curves. (English) Zbl 1267.14040
This paper deals with the question which algebraic curves can be defined over their field of moduli. Given an algebraic curve $$C$$, its field of moduli is the intersection of all fields over which $$C$$ can be defined; it is therefore the smallest possible field of definition that $$C$$ could allow, but a priori $$C$$ need not be defined over it. The main results in this paper are twofold.
The first result uses a signature argument applied to the canonical quotient map $$C \to C / \text{Aut}(C)$$. By using methods in [P. Débes and M. Emsalem, J. Algebra 211, 42-56 (1999; Zbl 0934.14019)], it is shown that $$C$$ is defined over its field of moduli if $$C / \text{Aut}(C)$$ has genus 0 and additionally one of the branch indices for this quotient map appears an odd number of times.
This first result is applied to cyclic $$q$$-gonal (and in particular hyperelliptic) curves. It is used to show that if the automorphism group of such a curve is large enough, then it can be defined over its field of moduli. Additionally, some new $$q$$-gonal curves whose field of moduli is contained in $$\mathbb{R}$$, but which cannot be defined over $$\mathbb{R}$$, are given. This generalizes results of B. Huggins [Math. Res. Lett. 14, No. 2, 249–262 (2007; Zbl 1126.14036)] for the hyperelliptic case.
If $$C$$ is a plane quartic, then the authors use the result to show that $$C$$ is defined over its field of moduli as long as its automorphism group is not cyclic of order two or the Klein Viergruppe. As a second main result of the paper, they use a Weil cocycle argument to show that in the latter case, any such quartic over $$\mathbb{C}$$ whose field of moduli is contained in $$\mathbb{R}$$ is defined over $$\mathbb{R}$$. Finally, the authors give an example of a plane quartic whose automorphism group is cyclic of order two, and which has field of moduli contained in $$\mathbb{R}$$, while it cannot be defined over $$\mathbb{R}$$.

##### MSC:
 14H37 Automorphisms of curves 14H10 Families, moduli of curves (algebraic) 14H45 Special algebraic curves and curves of low genus
##### Keywords:
algebraic curves; field of moduli; field of definition
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##### References:
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