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, Tome 30 (2018) no. 3, pp. 801-828.

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.

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.

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DOI : 10.5802/jtnb.1050
Classification : 11G05, 11G07, 14H52
Mots clés : 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.
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Tom Fisher; Lazar Radičević. Some minimisation algorithms  in arithmetic invariant theory. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 801-828. doi : 10.5802/jtnb.1050. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1050/

[1] Manjul Bhargava; Wei Ho Coregular spaces and genus one curves, Camb. J. Math., Volume 4 (2016) no. 1, pp. 1-119 | DOI | MR | Zbl

[2] Bryan John Birch; Henry Peter F. Swinnerton-Dyer Notes on elliptic curves I, J. Reine Angew. Math., Volume 212 (1963), pp. 7-25 | MR | Zbl

[3] John E. Cremona; Tom A. Fisher; Michael Stoll Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, Algebra Number Theory, Volume 4 (2010) no. 6, pp. 763-820 | DOI | MR | Zbl

[4] Tom A. Fisher Minimisation and reduction of 5-coverings of elliptic curves, Algebra Number Theory, Volume 7 (2013) no. 5, pp. 1179-1205 | DOI | MR | Zbl

[5] Tom A. Fisher; Rachel Newton Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve, Int. J. Number Theory, Volume 10 (2014) no. 7, pp. 1881-1907 | DOI | MR | Zbl

[6] Joseph H. Silverman The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 1986, xii+400 pages | MR | Zbl

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