Constructions arising from Néron’s high rank curves

Abstract:

Many papers quote Néron’s geometric construction of elliptic curves of rank $11$ over $\mathbb {Q}\;[{\mathbf {N}}]$—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 [${\mathbf {PP}},{\mathbf {GZ}},{\mathbf {Na}}$ and ${\mathbf {BK}}$] is to make $[{\mathbf {N}}]$ explicit. Excluding $[{\mathbf {BK}}]$, however, these papers derive little from Néron’s constructions. All show some lack of confidence in the details of $[{\mathbf {N}}]$. The core of this paper ($\S 3$), meets objections to $[{\mathbf {N}}]$ raised by correspondents. Our method adds a novelty as it magnifies the constructions of $[{\mathbf {N}}]$—"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 $11$ 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 $[{\mathbf {N}}]$. In particular $(\S 4(c))$, the explicit form of the irreducibility theorem in $[{\mathbf {Fr}}]$ applies to give explicit rank $11$ curves over $\mathbb {Q}$: with Selmer’s conjecture, rank $12$.