A geometric approach to the Cohen-Lenstra heuristics
Aaron Landesman1
1 Simons Building (Building 2), Room 106 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 947-997.

Nous donnons une nouvelle description géométrique du fait qu’un élément du groupe de classes d’un corps quadratique, vu comme une forme quadratique q, soit de n-torsion. Nous montrons que q correspond à un élément de n-torsion si et seulement s’il existe un polynôme de degré n dont le résultant avec q est ±1. Ceci est motivé par une paramétrisation géométrique plus précise, qui donne un lien direct entre la torsion dans les groupes de classes de corps quadratiques et certains groupes de Selmer de courbes de genre 1 singulières.

We give a new geometric description of when an element of the class group of a quadratic field, thought of as a quadratic form q, is n-torsion. We show that q corresponds to an n-torsion element if and only if there exists a degree n polynomial whose resultant with q is ±1. This is motivated by a more precise geometric parameterization which directly connects torsion in class groups of quadratic fields to Selmer groups of singular genus 1 curves.

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Révisé le :
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DOI : 10.5802/jtnb.1270
Classification : 11R29, 11R11
Mots clés : Cohen-Lenstra heuristics, geometric parameterization, class groups, quadratic fields
Aaron Landesman 1

1 Simons Building (Building 2), Room 106 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Aaron Landesman. A geometric approach to the Cohen-Lenstra heuristics. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 947-997. doi : 10.5802/jtnb.1270. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1270/

[1] Michael Artin; H. Peter F. Swinnerton-Dyer The Shafarevich–Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math., Volume 20 (1973), pp. 249-266 | DOI | MR | Zbl

[2] Arnaud Beauville Complex algebraic surfaces, London Mathematical Society Student Texts, 34, Cambridge University Press, 1996, x+132 pages (translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid) | DOI | MR

[3] Manjul Bhargava Higher composition laws. I. A new view on Gauss composition, and quadratic generalizations, Ann. Math., Volume 159 (2004) no. 1, pp. 217-250 | DOI | MR | Zbl

[4] Manjul Bhargava; Daniel M. Kane; Hendrik W. Lenstra; Bjorn Poonen; Eric Rains Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves, Camb. J. Math., Volume 3 (2015) no. 3, pp. 275-321 | DOI | MR | Zbl

[5] Manjul Bhargava; Arul Shankar; Xiaoheng Wang Squarefree values of polynomial discriminants I (2016) | arXiv

[6] Manjul Bhargava; Ila Varma The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders, Proc. Lond. Math. Soc., Volume 112 (2016) no. 2, pp. 235-266 | DOI | MR | Zbl

[7] Siegfried Bosch; Werner Lütkebohmert; Michel Raynaud Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 21, Springer, 1990, x+325 pages | DOI | MR

[8] Henri Cohen; Hendrik W. Lenstra Heuristics on class groups, Number Theory (Lecture Notes in Mathematics), Volume 1052, Springer, 1984, pp. 26-36 | DOI | MR | Zbl

[9] Henri Cohen; Hendrik W. Lenstra Heuristics on class groups of number fields, Number Theory Noordwijkerhout 1983 (Lecture Notes in Mathematics), Volume 1068, Springer, 1984, pp. 33-62 | DOI | MR | Zbl

[10] Jordan S. Ellenberg; Akshay Venkatesh; Craig Westerland Homological stability for Hurwitz spaces and the Cohen–Lenstra conjecture over function fields, Ann. Math., Volume 183 (2016) no. 3, pp. 729-786 | DOI | MR | Zbl

[11] Alexander Grothendieck Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas (Troisième partie), Publ. Math., Inst. Hautes Étud. Sci. (1966) no. 28, pp. 1-255 | MR | Zbl

[12] Alexander Grothendieck Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas (Quatrième partie), Publ. Math. IHES, Volume 32 (1967), pp. 1-361 | Zbl

[13] Alexander Grothendieck Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses, Dix exposés sur la cohomologie des schémas (Advanced Studies in Pure Mathematics), Volume 3, North-Holland, 1968, pp. 46-66 | MR | Zbl

[14] Aaron Landesman A thesis of minimal degree: two, Ph. D. Thesis, Stanford University, USA (2021)

[15] Aaron Landesman The Torelli map restricted to the hyperelliptic locus, Trans. Amer. Math. Soc., Ser. B, Volume 8 (2021), pp. 354-378 | DOI | MR | Zbl

[16] Serge Lang Algebra, Graduate Texts in Mathematics, 211, Springer, 2002, xvi+914 pages | DOI

[17] Bjorn Poonen Rational points on varieties, Graduate Studies in Mathematics, 186, American Mathematical Society, 2017, xv+337 pages | DOI | MR

[18] Bjorn Poonen; Eric Rains Random maximal isotropic subspaces and Selmer groups, J. Am. Math. Soc., Volume 25 (2012) no. 1, pp. 245-269 | DOI | MR | Zbl

[19] The Stacks Project Authors Stacks Project, http://stacks.math.columbia.edu

[20] Melanie Matchett Wood Gauss composition over an arbitrary base, Adv. Math., Volume 226 (2011) no. 2, pp. 1756-1771 | DOI | MR | Zbl

[21] Melanie Matchett Wood Parametrizing quartic algebras over an arbitrary base, Algebra Number Theory, Volume 5 (2011) no. 8, pp. 1069-1094 | DOI | MR | Zbl

[22] Melanie Matchett Wood Parametrization of ideal classes in rings associated to binary forms, J. Reine Angew. Math., Volume 689 (2014), pp. 169-199 | DOI | MR | Zbl

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