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On the rank of elliptic curves arising from Pythagorean quadruplets. II. (English) Zbl 1467.14075

Let \((a,b,c,d)\in \mathbb{Z}^4\) be a Pythagorean quadruplet, i.e. a (nontrivial) solution of \(a^2+b^2=c^2+d^2\) and define curves \[ E_{a,b,c,0}\ : y^2=(x-a^2)(x-b^2)(x-c^2)+a^2b^2 \,, \] \[ E_{a,b,c,1}\ : y^2=(x-a^2)(x-b^2)(x-c^2)+a^2b^2c^2 \,. \] The paper deals with the rank of subfamilies of curves of type \(E_{a,b,c,0}\) and \(E_{a,b,c,1}\). The authors use a parametrization of the Pythagorean quadruplets (namely, in non-homogenous coordinates, \((a,b,c,d)=(uv-1,u+v,uv+1,v-u)\)) to find non-trivial (i.e. different from \((a^2,ab)\), \((b^2,ab)\) in \(E_{a,b,c,0}\) and from \((a^2,abc)\), \((b^2,abc)\) in \(E_{a,b,c,1}\)) rational points on the curves by choosing an \(x\)-coordinate and then imposing conditions on \(u\) and \(v\) to get a square out of \((x-a^2)(x-b^2)(x-c^2)+a^2b^2\) or \((x-a^2)(x-b^2)(x-c^2)+a^2b^2c^2\). Via the specialization theorem and computations using SAGE and/or mwrank, they are able to check independence of these points and find families of elliptic curves of rank \(\geqslant 3\) inside \(E_{a,b,c,1}\) and of rank \(\geqslant 4\) inside \(E_{a,b,c,0}\), with a few examples of curves of rank 7 and 8 for the latter family.

MSC:

14H52 Elliptic curves
11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations

Citations:

Zbl 1441.14108
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References:

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