On Functional Equations of Dirichlet Functions

Lavrik, A. F.

doi:10.1070/im1967v001n02abeh000565

Cited by 20 publications

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“…One can similarly best-fit to e c D a (log D) b , but here the data seem 29 Except for d = 301980419090843394 (where we only have 5 independent points), the isogeny invariance of the BSD-quotient implies that none of the rank 6 curves has any 8-torsion in X. 30 Predictions along these lines (first seen in [11,Conjecture 2]), are sometimes called the "ratios conjecture" though that phrase has now taken upon a different meaning in the field of number theoretical random matrix theory, so we prefer "ratios prediction" instead. …”

mentioning

confidence: 99%

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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mentioning

confidence: 99%

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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“…The higher dimensional incomplete gamma functions of one complex variable have been shown by Lavrik [23] to arise whenever one has a Dirichlet series with functional equations involving multiple gamma factors. Such Dirichlet series include Art in L-f unctions of number fields.…”

Section: Jomentioning

confidence: 99%

“…[5, p. 207], [12], [20], and [24,Ch. 10] [3], [22], [24], [39], [40]), but it does not appear that anyone has obtained an explicit formula in the spirit of [23], except in the special case q = 2 in [2]. Instead of one functional equation, there are many.…”

Section: Jomentioning

confidence: 99%

“…This is the result needed for the analytic continuation of Eisenstein series as sums of such incomplete gamma functions. Lavrik has obtained in [23] expansions of Dirichlet series with functional equations involving multiple gamma factors of one complex variable, assuming that the analytic continuation and functional equation have already been proved by some other method. Lavrik writes his functions as inverse Mellin transforms.…”

Section: Jamentioning

confidence: 99%

“…To do the analytic continuation it is natural to attempt to write the Eisenstein series as sums of higher dimensional incomplete gamma functions, imitating the method of Riemann which gives formula (2.5) (cf. [23]). The motivation for using this method rather than the Maass-Selberg relations or existential arguments from several complex variables (cf.…”

Section: Jamentioning

confidence: 99%

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