Ranks of quadratic twists of elliptic curves

Abstract

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.

Heron quadrilaterals via elliptic curves

“…All of these curves arise from cyclic quadrilaterals. Furthermore, in the special case with α = 0, we find infinite families of congruent number curves with ranks 2 and 3, matching the results of [43], [51], [30].…”

“…In the remainder of this section, we give infinite families of congruent number elliptic curves with (at least) rank three. Searching for families of congruent curves with high rank has been done before [18], [19], [42], [43], [51]. Currently, the best known results are a few infinite families with rank at least 3 [30], [43], and several individual curves with rank 7 [51].…”

Elliptic Curves with Long Arithmetic Progressions Have Large Rank

“…with κ(j 0 , d) and C(j 0 , d) as defined in (6.3) and (6.5). Here we used that for j 0 fixed, one can check that the quantity C(j 0 , d) is increasing on d. We compute (6.6) log C(j 0 , 3) = 80 46 (log(c(j 0 , 3)) + log(6320) + 158 log (3)) < 80 46 (log(c(j 0 , 3)) + 183)…”

“…It is an open problem whether the ranks of elliptic curves over Q are uniformly bounded. Various heuristics have been developed in support of uniform boundedness [1,27,31,32,37,46]. Also, the second author has shown [28] that a conjecture of Lang in diophantine approximation implies uniform boundedness of ranks for families of elliptic curves with a fixed j-invariant.…”

New Rank Records For Elliptic Curves Having Rational Torsion

“…One side, claiming that ranks are bounded, has recently been bolstered by several different models [29,30,25] that predict that all but finitely many elliptic curves have rank at most 21, with stronger conjectured bounds on which ranks occur infinitely often for each possible torsion group T . (For example, if T = Z/nZ for n = 2, 3, .…”