eprintid: 10038649
rev_number: 38
eprint_status: archive
userid: 608
dir: disk0/10/03/86/49
datestamp: 2018-03-09 15:24:38
lastmod: 2020-08-01 23:44:42
status_changed: 2018-03-12 09:26:39
type: article
metadata_visibility: show
creators_name: Watkins, M
creators_name: Donnelly, S
creators_name: Elkies, N
creators_name: Fisher, T
creators_name: Granville, A
creators_name: Rogers, N
title: Ranks of quadratic twists of elliptic curves
ispublished: pub
divisions: UCL
divisions: A01
divisions: B04
divisions: C06
note: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
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.
date: 2014-11-27
date_type: published
official_url: http://pmb.univ-fcomte.fr/2014_en.html
oa_status: green
full_text_type: pub
language: eng
primo: open
primo_central: open_green
article_type_text: Article
verified: verified_manual
elements_id: 1512212
lyricists_name: Granville, Andrew
lyricists_id: AGRAN98
actors_name: Granville, Andrew
actors_id: AGRAN98
actors_role: owner
full_text_status: public
publication: Publications mathématiques de Besançon: Algèbre et Théorie des Nombres
number: 2
pagerange: 63-98
citation:        Watkins, M;    Donnelly, S;    Elkies, N;    Fisher, T;    Granville, A;    Rogers, N;      (2014)    Ranks of quadratic twists of elliptic curves.                   Publications mathématiques de Besançon: Algèbre et Théorie des Nombres   (2)   pp. 63-98.          Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10038649/7/Watkins_et_all.pdf