Some minimisation algorithms in arithmetic invariant theory
Tom Fisher1; Lazar Radičević1
1 University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK.
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 801-828.

We extend the work of Cremona, Fisher and Stoll on minimising genus one curves of degrees 2,3,4,5, to some of the other representations associated to genus one curves, as studied by Bhargava and Ho. Specifically we describe algorithms for minimising bidegree (2,2)-forms, 3×3×3 cubes and 2×2×2×2 hypercubes. We also prove a theorem relating the minimal discriminant to that of the Jacobian elliptic curve.

Nous étendons le travail de Cremona, Fisher et Stoll sur la minimisation des courbes de genre 1 et de degrés 2, 3, 4, 5, à d’autres représentations associées aux courbes de genre 1 étudiées par Bhargava et Ho. Plus précisément nous donnons des algorithmes pour minimiser les formes de bidegré (2,2), les cubes 3×3×3 et les hypercubes 2×2×2×2. Nous démontrons également un théorème reliant le discriminant minimal à celui de la courbe elliptique jacobienne.

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Accepted:
Published online:
DOI: 10.5802/jtnb.1050
Classification: 11G05, 11G07, 14H52
Keywords: elliptic curves, genus one curves, minimisation, coregular representations
Tom Fisher 1; Lazar Radičević 1

1 University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK.
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Tom Fisher; Lazar Radičević. Some minimisation algorithms  in arithmetic invariant theory. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 801-828. doi : 10.5802/jtnb.1050. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1050/

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