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Genus two curves covering elliptic curves: a computational approach

Tanush Shaska

I-AIMS paper 2004-2 · Published in Lecture Notes Ser. Comput. 13, World Sci. (2005), 206–231 · 2005

A genus 2 curve C has an elliptic subcover if there exists a degree n maximal covering ψ: C → E to an elliptic curve E. Degree n elliptic subcovers occur in pairs (E, E′). The Jacobian J_C of C is isogenous of degree n² to the product E × E′. We say that J_C is (n,n)-split. The locus of C, denoted by ℒₙ, is an algebraic subvariety of the moduli space ℳ₂. The space ℒ₂ was studied by Shaska and Völklein and by Gaudry and Schost. The space ℒ₃ was studied earlier, where an algebraic description was given as a sublocus of ℳ₂. In this survey we give a brief description of the spaces ℒₙ for a general n and then focus on small n. We describe some of the computational details which were skipped in earlier work. Further we explicitly describe the relation between the elliptic subcovers E and E′. We have implemented most of these relations in computer programs which check easily whether a genus 2 curve has (2,2) or (3,3) split Jacobian. In each case the elliptic subcovers can be explicitly computed.

MSC classes: Primary 14H45; Secondary 14H30, 14H37, 14Q05

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