Minimisation and reduction of 2-, 3-and 4-coverings of elliptic curves

作者：Cremona John E^*; Fisher Tom A; Stoll Michael

来源：Algebra and Number Theory, 2010, 4(6): 763-820.

DOI：10.2140/ant.2010.4.763

• elliptic curves
• genus-one curves
• minimisation
• reduction
• descent

We consider models for genus-one curves of degree n for n = 2, 3 and 4, which arise in explicit n-descent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general number fields. Finally, for genus-one models defined over Q, we develop a theory of reduction and again give explicit algorithms for n = 2, 3 and 4.

• 出版日期2010