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Minimal models for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}6$-coverings of elliptic curves

Part of: Curves Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 August 2014

Tom Fisher
Tom Fisher*
Affiliation:
University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom email T.A.Fisher@dpmms.cam.ac.uk

Abstract

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In this paper we give a new formula for adding $2$-coverings and $3$-coverings of elliptic curves that avoids the need for any field extensions. We show that the $6$-coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.

MSC classification

Secondary: 11G05: Elliptic curves over global fields 14H52: Elliptic curves
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Special Issue A: Algorithmic Number Theory Symposium XI , 2014 , pp. 112 - 127
Copyright
© The Author 2014 

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