I‑AIMS Institute for Artificial Intelligence
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Hyperelliptic curves with extra involutions

Jaime Gutierrez, Tanush Shaska

I-AIMS paper 2003-3 · Published in LMS J. Comput. Math. 8 (2005), 102–115 · 2005

The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus ℒ_g of such genus g hyperelliptic curves is a g-dimensional subvariety of the moduli space of hyperelliptic curves ℋ_g. We discover a birational parametrization of ℒ_g via dihedral invariants and show how these invariants can be used to determine the field of moduli of points 𝔭 ∈ ℒ_g. We conjecture that for 𝔭 ∈ ℋ_g with |Aut(𝔭)| > 2 the field of moduli is a field of definition and prove this conjecture for any point 𝔭 ∈ ℒ_g such that the Klein 4-group is embedded in the reduced automorphism group of 𝔭. Further, for g = 3 we show that for every moduli point 𝔭 ∈ ℋ₃ such that |Aut(𝔭)| > 4, the field of moduli is a field of definition and provide a rational model of the curve over its field of moduli.

MSC classes: Primary 11Gxx, 11G30; Secondary 14D22, 14H45

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