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Elliptic subfields and automorphisms of genus 2 function fields

Tony Shaska·Helmut Voelklein
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Abstract

We study genus 2 function fields with elliptic subfields of degree 2. The locus $Ł_2$ of these fields is a 2-dimensional subvariety of the moduli space $\mathcal M_2$ of genus 2 fields. An equation for $Ł_2$ is already in the work of Clebsch and Bolza. We use a birational parameterization of $Ł_2$ by affine 2-space to study the relation between the j-invariants of the degree 2 elliptic subfields. This extends work of Geyer, Gaudry, Stichtenoth and others. We find a 1-dimensional family of genus 2 curves having exactly two isomorphic elliptic subfields of degree 2; this family is parameterized by the j-invariant of these subfields.

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Tony Shaska, Helmut Voelklein. 2001-07-19. Elliptic subfields and automorphisms of genus 2 function fields. https://arxiv.org/abs/math/0107142

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