Rocky Mountain Journal of Mathematics

Symmetric diophantine systems and families of elliptic curves of high rank

Ajai Choudhry

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Abstract

While there has been considerable interest in the problem of finding elliptic curves of high rank over $\mathbb {Q}$, very few parametrized families of elliptic curves of generic rank $\geq 8$ have been published. In this paper we use solutions of certain symmetric diophantine systems to construct a number of families of elliptic curves whose coefficients are given in terms of several arbitrary parameters and whose generic rank ranges from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the elliptic curve $\leq 13$. It is, however, expected that the parametrized families of elliptic curves obtained in this paper would yield examples of elliptic curves of much higher rank.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 5 (2019), 1419-1447.

Dates
First available in Project Euclid: 19 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1568880087

Digital Object Identifier
doi:10.1216/RMJ-2019-49-5-1419

Mathematical Reviews number (MathSciNet)
MR4010569

Zentralblatt MATH identifier
07113695

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11D25: Cubic and quartic equations 11D41: Higher degree equations; Fermat's equation

Keywords
elliptic curves of high rank symmetric diophantine systems

Citation

Choudhry, Ajai. Symmetric diophantine systems and families of elliptic curves of high rank. Rocky Mountain J. Math. 49 (2019), no. 5, 1419--1447. doi:10.1216/RMJ-2019-49-5-1419. https://projecteuclid.org/euclid.rmjm/1568880087


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