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Elliptic subfields and automorphisms of genus 2 function fields

Tanush Shaska, Helmut Völklein

I-AIMS paper 2000-2 · Published in Algebra, Arithmetic and Geometry with Applications (Abhyankar’s 70th birthday), Springer (2004), 703–723 · 2004

We study genus 2 function fields with elliptic subfields of degree 2. The locus ℒ₂ of these fields is a 2-dimensional subvariety of the moduli space ℳ₂ of genus 2 fields. An equation for ℒ₂ is already in the work of Clebsch and Bolza. We use a birational parametrization of ℒ₂ 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.

This paper is dedicated to Professor Shreeram Abhyankar on the occasion of his 70th birthday.

MSC classes: Primary 14H05; Secondary 14H10, 14H37, 14H52

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