Distribution of the 2-Selmer rank under twisting

Mark Watkins

doi:10.5802/pmb.46

Mark Watkins ¹

¹ Department of Mathematics, Carslaw Building, University of Sydney, NSW 2006, Australia

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2022), pp. 59-133.

Abstract
Résumé

Consider the elliptic curve $y^{2} = x^{3} - x$ (which is associated to congruent numbers). Heath-Brown described the $2$ -Selmer rank distribution of the quadratic twists of this curve over the set of odd squarefree integers, using methods involving computing the moments of the 2-Selmer rank via an analysis of quadratic residue symbols (these appear in a Monsky matrix).

Swinnerton-Dyer then investigated the distribution of the 2-Selmer rank under twisting for a wider family (elliptic curves with full 2-torsion and no 4-torsion), using a Markov chain analysis to show the expected distribution. However, his result used an unnatural ordering of the integers, namely by the number of prime factors. This was remedied by Kane, who showed the result under the natural ordering via methodology similar to Heath-Brown’s.

More recently, Smith (as part of a larger work) has shown the same result using Swinnerton-Dyer’s method, essentially showing that the input (involving quadratic residue symbols of the prime divisors of the twist factor) to his Markov chain analysis can be shown to have the expected equi-distribution under the natural ordering. We give an exposition of Smith’s work, with an explicit (and effective) error bound. We also discuss the related problem of the 4-rank of quadratic class groups, initially done by Fouvry and Klüners.

On considère la courbe elliptique $y^{2} = x^{3} - x$ (qui est associée au problème des nombres congruents). Heath-Brown a décrit la distribution du rang du groupe du 2-Selmer des tordues quadratiques de cette courbe par les entiers impairs sans facteurs carrés en utilisant des méthodes sur le calcul des moments des rangs des 2-Selmer par une analyse des symboles quadratiques (qui apparaissent dans une matrice de Monsky).

Swinnerton-Dyer a ensuite étudié la distribution du rang des 2-Selmer pour une famille de tordues quadratiques plus large (de courbes elliptiques avec la 2 torsion complète et sans 4-torsion), en utilisant des chaînes de Markov pour obtenir la valeur attendue distribution. Cependant, son résultat utilise un ordre non naturel des entiers, à savoir un ordre donné par le nombre de facteurs premiers. Ceci a été complété par Kane, qui a démontré le même résultat avec l’ordre naturel via une méthode similaire à celle de Heath-Brown.

Plus récemment, Smith (dans le cadre d’un travail plus général) a également montré le même résultat en utilisant la méthode de Swinnerton-Dyer, en établissant essentiellement que la donnée (impliquant des symboles de résidus quadratiques des diviseurs premiers l’entier tordant la courbe de départ) pour son analyse de la chaîne de Markov conduit à la valeur attendue d’équi-distribution sous l’ordre naturel. Nous présentons le travail de Smith en donnant une borne d’erreur explicite (et efficace). Nous discutons également du problème connexe du 4-rang des groupes de classes quadratiques, initialement réalisé par Fouvry et Klüners.

Received: 2020-07-13
Published online: 2022-08-18

DOI: 10.5802/pmb.46

Classification: 11G05, 11G40, 11L40
Keywords: congruent number problem, Selmer groups, ranks of elliptic curves, Legendre symbol distribution

Author's affiliations:

Mark Watkins ¹

¹ Department of Mathematics, Carslaw Building, University of Sydney, NSW 2006, Australia

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Mark Watkins. Distribution of the 2-Selmer rank under twisting. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2022), pp. 59-133. doi : 10.5802/pmb.46. https://pmb.centre-mersenne.org/articles/10.5802/pmb.46/

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