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On the field of moduli of superelliptic curves

Rubén A. Hidalgo, Tanush Shaska

I-AIMS paper 2016-1 · Published in Contemp. Math. 703 (2018), 47–62 · 2018

A superelliptic curve 𝒳 of genus g ≥ 2 is not necessarily defined over its field of moduli but it can be defined over a quadratic extension of it. While a lot of work has been done by many authors to determine which hyperelliptic curves are defined over their field of moduli, less is known for superelliptic curves. In this paper we observe that if the reduced group of a genus g ≥ 2 superelliptic curve 𝒳 is different from the trivial or cyclic group, then 𝒳 can be defined over its field of moduli; in the cyclic situation we provide a sufficient condition for this to happen. We also determine those families of superelliptic curves of genus at most 10 which might not be definable over their field of moduli.

MSC classes: Primary 30F10; Secondary 14H37, 14H45

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