Elliptic curve involving subfamilies of rank at least 5 over $\mathbb{Q}(t)$ or $\mathbb{Q}(t,k)$

Authors : Ahmed El Amine YOUMBAI, A Muhammed ULUDAĞ, Djilali BEHLOUL

Pages : 721-731

Doi:10.15672/hujms.708945

View : 25 | Download : 7

Publication Date : 2021-06-07

Article Type : Research Paper

Abstract :Motivated by the work of Zargar and Zamani, we introduce a family of elliptic curves containing several one- insert ignore into journalissuearticles values(respectively two-); parameter subfamilies of high rank over the function field $\mathbb{Q}insert ignore into journalissuearticles values(t);$ insert ignore into journalissuearticles values(respectively $\mathbb{Q}insert ignore into journalissuearticles values(t,k);$);. Following the approach of Moody, we construct two subfamilies of infinitely many elliptic curves of rank at least 5 over $\mathbb{Q}insert ignore into journalissuearticles values(t,k);$. Secondly, we deduce two other subfamilies of this family, induced by the edges of a rational cuboid containing five independent $\mathbb{Q}insert ignore into journalissuearticles values(t);$-rational points. Finally, we give a new subfamily induced by Diophantine triples with rank at least 5 over $\mathbb{Q}insert ignore into journalissuearticles values(t);$. By specialization, we obtain some specific examples of elliptic curves over $\mathbb{Q}$ with a high rank insert ignore into journalissuearticles values(8, 9, 10 and 11);.
Keywords : Elliptic Curves, Rank, Rational Cuboid, Diophantine Triples