On elliptic curves and random matrix theory

Watkins, Marley W.

doi:10.5802/jtnb.653

Cited by 3 publications

(3 citation statements)

References 31 publications

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“…Plugging into the above, we get r ≤ 2η + 3 2l as D → ∞, so in particular any l > 0 gives an upper bound on ranks in twist families. Contrarily, allowing l > 3/2 implies r ≤ 2 for the generic case η = 1, while the data of [58] suggest otherwise. Granville offers, in relation to the size of solutions to Pell equations, that l = 1 2 seems reasonable, leading to r ≤ 2η + 3.…”

Section: 2mentioning

confidence: 98%

Ranks of quadratic twists of elliptic curves

Watkins¹,

Donnelly²,

Elkies³

et al. 2015

Publications Mathématiques De Besançon. Algèbre Et Théorie Des Nombres

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A l g è b r e e t t h é o r i e d e s n o m b r e s Abstract. -We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find a rank 8 twist, and discuss how our results here compare to some predictions of rank growth vis-à-vis conductor. Finally we explicate a heuristic of Granville, which when interpreted judiciously could predict that 7 is indeed the maximal rank in this quadratic twist family.Résumé. -Nous donnons un compte rendu d'un projet de grande envergure sur les rangs des courbes elliptiques dans une famille de tordues quadratiques en nous focalisant sur les courbes associées aux nombres congruents. Afin d'exclure certaines courbes, nos méthodes incluent des tests sur les 2,4,8-groupes de Selmer, l'utilisation de la formule explicite de Guinand-Weil et également des 3-descentes dans quelques cas. Nous constatons que les tordues quadratiques de rang 6 sont assez répandues (bien que toujours assez difficile à trouver), alors que celles de rang 7 semblent bien plus rares. Nous décrivons aussi notre incapacité à obtenir des tordues quadratiques de rang 8 et expliquons en quoi nos résultats peuvent se comparer à certaines prédictions sur la croissance du rang en fonction du conducteur. Enfin, nous expliquons une heuristique due à Granville, qui, lorsqu'elle est interprétée judicieusement, pourrait prédire que le rang maximal pour cette famille est en effet égal à 7.

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Section: 2mentioning

confidence: 98%

Ranks of quadratic twists of elliptic curves

Watkins¹,

Donnelly²,

Elkies³

et al. 2015

Publications Mathématiques De Besançon. Algèbre Et Théorie Des Nombres

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“…(See also [Del05] for related conjectures on the regulators.) There is also numerical data [Elk02,DD03,Wat08b]. According to Rubin and Silverberg [RS02, p. 466], the numerical data of Elkies suggests that N ≥3,odd (D) is about D 3/4 .…”

Section: Notation and Conventionsmentioning

confidence: 99%

A heuristic for boundedness of ranks of elliptic curves

et al. 2019

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We present a heuristic that suggests that ranks of elliptic curves E over Q are bounded. In fact, it suggests that there are only finitely many E of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.Our model is inspired in part by the Cohen-Lenstra heuristics for class groups [CL84], as reinterpreted by Friedman and Washington [FW89]. These heuristics predict that for a fixed odd prime p, the distribution of the p-primary part of the class group of a varying imaginary quadratic field is equal to the limit as n → ∞ of the distribution of the cokernel of the homomorphism Z n p A → Z n p given by a random matrix A ∈ M n (Z p ); see Section 4 for the precise conjecture. In analogy, and in agreement with conjectures of Delaunay [Del01, Del07, DJ14], Bhargava, Kane, Lenstra, Poonen, and Rains [BKLPR15] predicted that for a fixed prime p and r ∈ Z ≥0 , the distribution of the p-primary part of the Shafarevich-Tate group X(E) as E varies over rank r elliptic curves over Q ordered by height equals the limit as n → ∞ (through integers of the same parity as r) of the distribution of coker A for a random alternating matrix A ∈ M n (Z p ) subject to the condition rk Zp (ker A) = r; see Section 5 for the precise conjecture and the evidence for it.If imposing the condition rk Zp (ker A) = r yields a distribution conjecturally associated to the curves of rank r, then naturally we guess that if we choose A at random from the space M n (Z p ) alt of all alternating matrices without imposing such a condition, then the distribution of rk Zp (ker A) tends as n → ∞ to the distribution of the rank of an elliptic curve. This cannot be quite right, however: since an alternating matrix always has even rank, the parity of n dictates the parity of rk Zp (ker A). But if we choose n uniformly at random from {⌈η⌉, ⌈η⌉ + 1} (with η → ∞), then we find that rk Zp (ker A) equals 0 or 1 with probability 50% each, and rk Zp (ker A) ≥ 2 with probability 0%; for example, when n is even, we have rk Zp (ker A) = 0 unless det A = 0, and det A = 0 holds only when A lies on a (measure 0) hypersurface in the space M n (Z p ) alt of all alternating matrices. This 50%-50%-0% conclusion matches the elliptic curve rank behavior conjectured for quadratic twist families by Goldfeld [Gol79, Conjecture B] and Katz and Sarnak [KS99a, KS99b].So far, however, this model does not predict anything about the number of curves of each rank ≥ 2 except to say that asymptotically they should amount to 0% of curves. Instead of sampling from M n (Z p ), we could sample from the set M n (Z) alt,≤X of alternating integer matrices whose entries have absolute values bounded by X, and studybut this again would be 0 for each r ≥ 2. To obtain finer information, instead of taking the limit as X → ∞, we let X depend on the height H of the elliptic curve bein...

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“…Consider first the case F = Q and weight k. We follow the heuristic [4,19] prescribed by random matrix theory, verified by the large scale computation of central values of L-functions twisted by a quadratic character over Q; see also further work by Watkins [44]. In this theory, low-lying zeros of L-functions are related to values of characteristic polynomials of random matrices of SO(2m) and we deduce…”

Section: The Power Of Logmentioning

confidence: 99%