## Rocky Mountain Journal of Mathematics

### Symmetric diophantine systems and families of elliptic curves of high rank

Ajai Choudhry

#### 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

#### 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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