Third Italian Number Theory Meeting

Linear relations on families of powers of elliptic curves

speaker: Fabrizio Barroero (University of Manchester)

abstract: In a recent work Masser and Zannier showed that there are at most finitely many complex numbers \( \lambda \neq 0,1 \) such that the points \((2, \sqrt{2(2-\lambda)})\) and \((3, \sqrt{6(3-\lambda)})\) are simultaneously torsion on the Legendre elliptic curve \(E_\lambda\) of equation \(y^2=x(x-1)(x-\lambda)\). This is a special case of conjectures about Unlikely Intersections on families of abelian varieties, proved later in the two dimensional case by the same authors. As a natural higher dimensional extension, we considered the case of three points \((2, \sqrt{2(2-\lambda)})\), \((3, \sqrt{6(3-\lambda)})\) and \((5, \sqrt{20(5-\lambda)})\) and proved that there are at most finitely many \(\lambda \neq 0,1\) such that these three points satisfy two independent linear relations on \(E_\lambda\). This is a special case of a more general result in the framework of the conjectures mentioned above.

timetable:
Tue 22 Sep, 14:30 - 14:55, Aula Contini
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