On binary cubic and quartic forms
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 323-341.

In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to give precise asymptotic formulae for the number of integers in an interval representable by a binary cubic or quartic form and extends work of Hooley. Further, we give the field of definition of lines contained in certain cubic and quartic surfaces related to binary cubic and quartic forms.

Dans cet article, nous décrivons le groupe d’automorphismes rationnels d’une forme binaire cubique ou quartique à coefficients entiers et à discriminant non nul en termes de certains covariants quadratiques des formes cubiques et quartiques. Cela nous permet d’étendre les travaux de Hooley et de donner des formules asymptotiques précises pour le nombre d’entiers appartenant à un intervalle et représentables par une forme cubique ou quartique donnée. En outre, nous déterminons le corps de définition des droites contenues dans certaines surfaces cubiques et quartiques associées à des formes cubiques et quartiques binaires.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1083
Classification: 11D45,  11E76,  11D25
Keywords: Binary forms, cubic and quartic surfaces
Stanley Yao Xiao 1

1 Department of Mathematics University of Toronto Bahen Centre 40 St. George Street, Room 6290 Toronto, Ontario, Canada M5S 2E4
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Stanley Yao Xiao. On binary cubic and quartic forms. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 323-341. doi : 10.5802/jtnb.1083. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1083/

[1] Michael A. Bean The practical computation of areas associated with binary quartic forms, Math. Comput., Volume 66 (1997) no. 219, pp. 1269-1293 | Article | MR: 1397439 | Zbl: 0886.65153

[2] Michael A. Bennett; Sander R. Dahmen Klein forms and the generalized superelliptic equation, Ann. Math., Volume 177 (2013) no. 1, pp. 177-239 | MR: 2999040 | Zbl: 1321.11059

[3] Samuel Boissière; Alessandra Sarti Counting lines on surfaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 6 (2007) no. 1, pp. 39-52 | MR: 2341513 | Zbl: 1150.14013

[4] John E. Cremona Reduction of binary cubic and quartic forms, LMS J. Comput. Math., Volume 2 (1999), pp. 64-94 | MR: 1693411 | Zbl: 0927.11020

[5] Torsten Ekedahl An effective version of Hilbert’s irreducibility theorem, Séminaire de théorie des nombres (Paris, 1988-1989) (Progress in Mathematics), Volume 91, Birkhäuser, 1988, pp. 241-249 | Zbl: 0729.12005

[6] David R. Heath-Brown The density of rational points on curves and surfaces, Ann. Math., Volume 155 (2002), pp. 553-598 | Article | MR: 1906595 | Zbl: 1039.11044

[7] Christopher Hooley On binary cubic forms, J. Reine Angew. Math., Volume 226 (1967), pp. 30-87 | MR: 213299 | Zbl: 0163.04605

[8] Christopher Hooley On binary quartic forms, J. Reine Angew. Math., Volume 366 (1986), pp. 32-52 | MR: 833012 | Zbl: 0577.10022

[9] Christopher Hooley On binary cubic forms. II, J. Reine Angew. Math., Volume 521 (2000), pp. 185-240 | MR: 1752300 | Zbl: 0989.11019

[10] Gaston Julia Étude sur les formes binaires non quadratiques à indéterminées réelles ou complexes, Mem. Acad. Scin. l’Inst. France, Volume 55 (1917), pp. 1-293

[11] Felix Klein Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Dover Publications, 1956 | Zbl: 0072.25901

[12] Peter J. Olver Classical Invariant Theory, London Mathematical Society Student Texts, 44, Cambridge University Press, 1999 | MR: 1694364 | Zbl: 0971.13004

[13] Per Salberger Counting rational points on projective varieties (2009) (preprint)

[14] Beniamino Segre On arithmetical properties of quartic surfaces, Proc. Lond. Math. Soc., Volume 49 (1946), pp. 353-395 | Article | MR: 21952 | Zbl: 0034.08603

[15] Cameron L. Stewart; Stanley Yao Xiao On the representation of integers by binary forms, Math. Ann., Volume 375 (2019) no. 1-2, pp. 133-163 | Article | MR: 4000237 | Zbl: 07104238

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