Constructions arising from Néron’s high rank curves

Abstract

Many papers quote Néron’s geometric construction of elliptic curves of rank 11 11 over Q [ N ] \mathbb {Q}\;[{\mathbf {N}}] —still, at the writing of this paper, the elliptic curves of highest demonstrated rank. The purported reason for the ordered display of "creeping rank" in [ P P , G Z , N a {\mathbf {PP}},{\mathbf {GZ}},{\mathbf {Na}} and B K {\mathbf {BK}} ] is to make [ N ] [{\mathbf {N}}] explicit. Excluding [ B K ] [{\mathbf {BK}}] , however, these papers derive little from Néron’s constructions. All show some lack of confidence in the details of [ N ] [{\mathbf {N}}] . The core of this paper ( § 3 \S 3 ), meets objections to [ N ] [{\mathbf {N}}] raised by correspondents. Our method adds a novelty as it magnifies the constructions of [ N ] [{\mathbf {N}}] —"generation of pencils of cubics from their singular fibers". This has two advantages: it displays (Remark 4.2) the free parameters whose specializations give high rank curves; and it demonstrates the existence of rank 11 11 curves through one appeal only to Hilbert’s irreducibility theorem. That is, we have eliminated the unusual analogue of Hilbert’s result that takes up most of [ N ] [{\mathbf {N}}] . In particular ( § 4 ( c ) ) (\S 4(c)) , the explicit form of the irreducibility theorem in [ F r ] [{\mathbf {Fr}}] applies to give explicit rank 11 11 curves over Q \mathbb {Q} : with Selmer’s conjecture, rank 12 12 .