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Genus 2 fields with degree 3 elliptic subfields. (English) Zbl 1098.14017
Let \(k\) be an algebraically closed field of characteristic 0. The author studies the locus \(\mathcal{L}_3\) in the moduli space of genus 2 curves over \(k\) consisting of those curves which have a degree 3 map to an elliptic curve. A genus 2 curve \(C/k\) has a defining equation \(Y^2 = f(X,Z)\), where \(f\) is a sextic form over \(k\), determined up to the action of \( \text{ GL}_2(k)\) on the coordinates. There is a complete set of classical invariants for such forms, \(J_2\), \(J_4\), \(J_6\), and \(J_{10}\). The author gives an explicit (and large) defining equation for \(\mathcal{L}_3\) in terms of these invariants. Further, he shows that for any \(C \in \mathcal{L}_3\), the number of \(\text{ Aut}_k(C)\)-orbits of degree 3 elliptic subcovers is 1, 2, or 4. There are exactly two genus 2 curves for which the number is 4, a two-dimensional family for which it is 2, and a one-dimensional family for which it is 1. Finally, the author constructs a 2-parameter family of genus two curves which yields an explicit degree 2 cover from \(k^2\) minus a discriminant locus to a Zariski open subset of \(\mathcal{L}_3\), and gives an equation for the branch locus of this cover.

14H05 Algebraic functions and function fields in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14H45 Special algebraic curves and curves of low genus
11G99 Arithmetic algebraic geometry (Diophantine geometry)
Full Text: DOI
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