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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.

MSC:
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)
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