Shamsi Zargar, Arman On the rank of elliptic curves arising from Pythagorean quadruplets. (English) Zbl 1441.14108 Kodai Math. J. 43, No. 1, 129-142 (2020). Let \(a,b,c,d\in \mathbb{Z}\) form a Pythagorean quadruplet, i.e. they verify \(a^2+b^2=c^2+d^2\) and consider the associated curve \(y^2=(x-a^2)(x-b^2)(x-c^2)\) which, when \(|a|\neq|b|\neq |c|\) is an elliptic curve with full 2-torsion over \(\mathbb{Q}\). The paper deals with the rank of curves of this kind: using a non-homogenous form for Pythagorean quadruplets \((a,b,c,d)=(uv-1,u+v,uv+1,v-u)\), the author exhibits the rational point \(P_1=((u+v)^2,(uv-1)(u-v)(u+v))\) and then looks for values of \(u\) providing other independent rational points. At first the paper gives families of curves \(E_{u,v}\) (with \(u=f(v,m)\) depending on a parameter \(m\)) with rank 2, then, for some \(m\) of type \(\frac{4(w^2-2)}{w^2+2}\), the author is able to show (sub)families of rank 3 and 4. Most proofs rely on Silverman’s specialization theorem and on an algorithm of I. Gusić and P. Tadić [Glas. Mat., III. Ser. 47, No. 2, 265–275 (2012; Zbl 1300.11060)] exploiting the presence of the full 2-torsion: the use of the SAGE and MWRANK programs allows the author to check ranks of specializations and also to provide some examples of curves in the families above with rank 6 and 8. Reviewer: Andrea Bandini (Pisa) Cited in 1 ReviewCited in 4 Documents MSC: 11G05 Elliptic curves over global fields 11D25 Cubic and quartic Diophantine equations 11D09 Quadratic and bilinear Diophantine equations 14H52 Elliptic curves Keywords:elliptic curve; independent point; \(K3\) surface; Pythagorean quadruplet; rank Citations:Zbl 1300.11060 PDFBibTeX XMLCite \textit{A. Shamsi Zargar}, Kodai Math. J. 43, No. 1, 129--142 (2020; Zbl 1441.14108) Full Text: DOI Euclid