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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Constructions arising from Néron's high rank curves

Author: M. Fried
Journal: Trans. Amer. Math. Soc. 281 (1984), 615-631
MSC: Primary 14K07; Secondary 11G05, 14G25, 14K15
MathSciNet review: 722765
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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 ($ \S3$), 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 $ (\S4(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$.

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Article copyright: © Copyright 1984 American Mathematical Society

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