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Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves. (English) Zbl 1222.11073

Summary: 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 \(\mathbb Q\), we develop a theory of reduction and again give explicit algorithms for \(n = 2\), 3 and 4.

MSC:

11G05 Elliptic curves over global fields
11G07 Elliptic curves over local fields
14H52 Elliptic curves
14H25 Arithmetic ground fields for curves
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