Some minimisation algorithms in arithmetic invariant theory
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 : https://doi.org/10.5802/jtnb.1050
Classification : 11G05,  11G07,  14H52
Mots clés : elliptic curves, genus one curves, minimisation, coregular representations
@article{JTNB_2018__30_3_801_0,
     author = {Tom Fisher and Lazar Radi\v{c}evi\'c},
     title = {Some minimisation algorithms  in arithmetic invariant theory},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {801--828},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1050},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1050/}
}
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/

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