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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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