Distribution of the 2-Selmer rank under twisting

Mark Watkins

doi:10.5802/pmb.46

Distribution of the 2-Selmer rank under twisting
[Distribution des rangs des 2 groupes de Selmer pour les tordues quadratiques]

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.

Résumé
Abstract

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.

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.

Reçu le : 2020-07-13
Publié le : 2022-08-18

DOI : 10.5802/pmb.46

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

Affiliations des auteurs :

Mark Watkins ¹

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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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/

Bibliographie
Cité par

[1] Arthur O. L. Atkin; Joseph Lehner Hecke operators on $Γ_{0} (m)$ , Math. Ann., Volume 185 (1970) no. 2, pp. 134-160 | DOI | Zbl

[2] 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

[3] Stephanie Chan; Peter Koymans; Djordjo Milovic; Carlo Pagano On the negative Pell equation (2019) (https://arxiv.org/abs/1908.01752)

[4] 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

[5] Christophe Delaunay Heuristics on Tate–Shafarevitch groups of elliptic curves defined over $ℚ$ , Exp. Math., Volume 10 (2001) no. 2, pp. 191-196 | DOI | MR | Zbl

[6] Pál Erdős; Mark Kac The Gaussian law of errors in the theory of additive number theoretic functions, Am. J. Math., Volume 62 (1940) no. 1, pp. 738-742 | DOI | MR | Zbl

[7] Étienne Fouvry; Jürgen Klüners Cohen–Lenstra heuristics of quadratic number fields, Algorithmic Number Theory (Lecture Notes in Computer Science), Volume 4076, Springer, 2006, pp. 40-55 | DOI | MR | Zbl

[8] Étienne Fouvry; Jürgen Klüners On the 4-rank of class groups of quadratic number fields, Invent. Math., Volume 167 (2007), pp. 455-513 | DOI | MR | Zbl

[9] Étienne Fouvry; Jürgen Klüners On the negative Pell equation, Ann. Math., Volume 172 (2010) no. 3, pp. 2035-2104 | DOI | MR | Zbl

[10] Étienne Fouvry; Jürgen Klüners The parity of the period of the continued fraction of $\sqrt{d}$ , Proc. Lond. Math. Soc., Volume 101 (2010) no. 2, pp. 337-391 | DOI | MR | Zbl

[11] John Friedlander; Henryk Iwaniec The polynomial $X^{2} + Y^{4}$ captures its primes, Ann. Math., Volume 148 (1998) no. 3, pp. 945-1040 | DOI | MR | Zbl

[12] Frank Gerth The 4-class ranks of quadratic fields, Invent. Math., Volume 77 (1984), pp. 489-515 | DOI | MR | Zbl

[13] Frank Gerth; Sidney W. Graham Application of a character sum estimate to a 2-class number density, J. Number Theory, Volume 19 (1984), pp. 239-247 | DOI | MR | Zbl

[14] Dorian M. Goldfeld The class number of quadratic fields and the conjectures of Birch and Swinnerton–Dyer, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 3 (1976) no. 4, pp. 624-663 | MR | Zbl

[15] Dorian M. Goldfeld; Andrzej Schinzel On Siegel’s zero, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 2 (1976) no. 4, pp. 571-583 | MR | Zbl

[16] Benedict H. Gross; Don B. Zagier Heegner points and derivatives of $L$ -series, Invent. Math., Volume 84 (1986) no. 2, pp. 225-320 | DOI | MR | Zbl

[17] D. Roger Heath-Brown The size of Selmer groups for the congruent number problem, Invent. Math., Volume 111 (1993) no. 1, pp. 171-195 | DOI | MR | Zbl

[18] D. Roger Heath-Brown The size of Selmer groups for the congruent number problem, II, Invent. Math., Volume 118 (1994) no. 2, pp. 331-370 | DOI | MR | Zbl

[19] D. Roger Heath-Brown A mean value estimate for real character sums, Acta Arith., Volume 72 (1995) no. 3, pp. 235-275 | DOI | MR | Zbl

[20] Hans Heilbronn On the averages of some arithmetical functions of two variables, Mathematika, Volume 5 (1958), pp. 1-7 | DOI | MR | Zbl

[21] Hsien-Kuei Hwang Sur la répartition des valeurs des fonctions arithmétiques. Le nombre de facteurs premiers d’un entier, J. Number Theory, Volume 69 (1998) no. 2, pp. 135-152 | DOI | MR | Zbl

[22] Matti Jutila On mean values of Dirichlet polynomials with real characters, Acta Arith., Volume 27 (1975), pp. 191-198 | DOI | MR | Zbl

[23] Daniel M. Kane On the ranks of the $2$ -Selmer groups of twists of a given elliptic curve, Algebra Number Theory, Volume 7 (2013) no. 5, pp. 1253-1279 | DOI | MR | Zbl

[24] Zev Klagsbrun; Barry Mazur; Karl Rubin Disparity in Selmer ranks of quadratic twists of elliptic curves, Ann. Math., Volume 178 (2013) no. 1, pp. 1-34 | MR | Zbl

[25] Zev Klagsbrun; Barry Mazur; Karl Rubin A Markov model for Selmer ranks in families of twists, Compos. Math., Volume 150 (2014) no. 7, pp. 1077-1106 | DOI | MR | Zbl

[26] Viktor A. Kolyvagin Finiteness of $E (ℚ)$ and $Ш (E, ℚ)$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 52 (1988) no. 3, pp. 522-540

[27] Peter Koymans; Djordjo Milovic On the 16-Rank of Class Groups of $ℚ (\sqrt{- 2 p})$ for Primes $p \equiv 1$ mod 4, Int. Math. Res. Not., Volume 2019 (2019) no. 23, pp. 7406-7427 | DOI | Zbl

[28] Peter Koymans; Carlo Pagano Effective convergence of coranks of random Rédei matrices (2020) (https://arxiv.org/abs/2005.12899)

[29] Kenneth Kramer Arithmetic of elliptic curves upon quadratic extension, Trans. Am. Math. Soc., Volume 264 (1981), pp. 121-135 | DOI | MR | Zbl

[30] Edmund Landau Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. der Math. u. Phys. (3), Volume 13 (1908), pp. 305-312 | Zbl

[31] Edmund Landau Über die Klassenzahl imaginär-quadratischer Zahlkörper, Gött. Nachr., Volume 1918 (1918), pp. 285-295 | Zbl

[32] Georg Landsberg Ueber eine Anzahlbestimmung und eine damit zusammenhängende Reihe, J. Reine Angew. Math., Volume 111 (1893), pp. 87-88 | MR | Zbl

[33] Qing Lu 8-rank of the class group and isotropy index, Sci. China, Math., Volume 58 (2015) no. 7, pp. 1433-1444 | MR | Zbl

[34] Barry Mazur; Karl Rubin Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math., Volume 181 (2010) no. 3, pp. 541-575 | DOI | MR | Zbl

[35] Franz Mertens Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math., Volume 78 (1874), pp. 46-62 | MR

[36] Paul Monsky Generalizing the Birch-Stephens theorem I. Modular curves, Math. Z., Volume 221 (1996) no. 3, pp. 415-420 | DOI | MR | Zbl

[37] Beresford N. Parlett; Christian Reinsch Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors, Numer. Math., Volume 13 (1969) no. 4, pp. 293-304 | DOI | Zbl

[38] Laslo Rédei Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper, J. Reine Angew. Math., Volume 171 (1934), pp. 55-60 | DOI | Zbl

[39] Laslo Rédei Über einige Mittelwertfragen im quadratischen Zahlkörper, J. Reine Angew. Math., Volume 174 (1936), pp. 15-55 | DOI

[40] Laslo Rédei Ein neues zahlentheoretisches Symbol mit Anwendungen auf die Theorie der quadratischen Zahlkörper, J. Reine Angew. Math., Volume 180 (1939), pp. 1-43 | DOI | Zbl

[41] Laslo Rédei; Hans Reichardt Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, J. Reine Angew. Math., Volume 170 (1934), pp. 69-74 | DOI | Zbl

[42] L. G. Sathe On a Problem of Hardy on The Distribution of Integers having a Given Number of Prime Factors I, J. Indian Math. Soc., Volume 17 (1953), pp. 63-82 | Zbl

[43] L. G. Sathe On a Problem of Hardy on The Distribution of Integers having a Given Number of Prime Factors II, J. Indian Math. Soc., Volume 17 (1953), pp. 83-141 | MR | Zbl

[44] L. G. Sathe On a Problem of Hardy on The Distribution of Integers having a Given Number of Prime Factors III, J. Indian Math. Soc., Volume 18 (1954), pp. 27-42 | MR | Zbl

[45] L. G. Sathe On a Problem of Hardy on The Distribution of Integers having a Given Number of Prime Factors IV, J. Indian Math. Soc., Volume 18 (1954), pp. 43-81 | MR | Zbl

[46] Atle Selberg Note on a paper by L. G. Sathe, J. Indian Math. Soc., Volume 18 (1954), pp. 83-87 | MR | Zbl

[47] Carl L. Siegel Über die Classenzahl quadratischer Zahlkörper, Acta Arith., Volume 1 (1935), pp. 83-86 | DOI | Zbl

[48] A. Smith $2^{\infty}$ -Selmer groups, $2^{\infty}$ -class groups, and Goldfeld’s conjecture (2017) (preprint)

[49] Peter Stevenhagen The Number of Real Quadratic Fields Having Units of Negative Norm, Exp. Math., Volume 2 (1993) no. 2, pp. 121-136 | DOI | MR | Zbl

[50] Peter Stevenhagen Rédei-matrices and applications, Number theory (Paris, 1992-1993) (London Mathematical Society Lecture Note Series), Volume 215, Cambridge University Press, 1995, pp. 245-259 | DOI | Zbl

[51] Peter Stevenhagen Redei reciprocity, governing fields, and negative Pell, Math. Proc. Camb. Philos. Soc., Volume 172 (2022) no. 3, pp. 627-654 | DOI | MR | Zbl

[52] Peter Swinnerton-Dyer 2-descent through the ages, Ranks of Elliptic Curves and Random Matrix Theory (London Mathematical Society Lecture Note Series), Volume 341, Cambridge University Press, 2007, pp. 345-356 | DOI | MR | Zbl

[53] Peter Swinnerton-Dyer The effect of twisting on the $2$ -Selmer group, Math. Proc. Camb. Philos. Soc., Volume 145 (2008) no. 3, pp. 513-526 | DOI | MR | Zbl

[54] Christian Tudesq Majoration de la loi locale de certaines fonctions additives, Arch. Math., Volume 67 (1996) no. 6, pp. 465-472 | DOI | MR | Zbl

[55] Pál Turán On a Theorem of Hardy and Ramanujan, J. Lond. Math. Soc., Volume 9 (1934), pp. 274-276 | DOI | MR | Zbl

[56] Arnold Walfisz Zur additiven Zahlentheorie. II, Math. Z., Volume 40 (1935), pp. 592-607 | DOI | MR | Zbl

[57] M. Watkins Notes on 4-Selmer and 8-class ranks (2020) (preprint)

[58] Maosheng Xiong On Selmer groups of quadratic twists of elliptic curves with a two-torsion over $ℚ$ , Mathematika, Volume 59 (2013) no. 2, pp. 303-319 | DOI | MR | Zbl

[59] Maosheng Xiong; Alexandru Zaharescu Distribution of Selmer groups of quadratic twists of a family of elliptic curves, Adv. Math., Volume 219 (2008) no. 2, pp. 523-553 | DOI | MR | Zbl

[60] Gang Yu Rank 0 quadratic twists of a family of elliptic curves, Compos. Math., Volume 135 (2003) no. 3, pp. 331-356 | MR | Zbl

[61] Gang Yu On the quadratic twists of a family of elliptic curves, Mathematika, Volume 52 (2005) no. 1-2, pp. 139-154 | MR | Zbl

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