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Hyperelliptic curves with reduced automorphism group A₅

David Sevilla, Tanush Shaska

I-AIMS paper 2005-2 · Published in Appl. Algebra Engrg. Comm. Comput. 18 (2007), 3–20 · 2007

We study genus g hyperelliptic curves with reduced automorphism group A₅ and give equations y² = f(x) for such curves in both cases where f(x) is a decomposable polynomial in x² or x⁵. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model y² = F(x) or y² = xF(x) of the curve over its field of moduli where F(x) can be chosen to be decomposable in x² or x⁵. While similar equations have been given over ℝ, this is the first time that these equations are given over the field of moduli of the curve.

MSC classes: Primary 14H37; Secondary 14H25, 14H10

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