U utorak, 19.04.2022. akademik prof. dr. sc. Andrej Dujella održat će predavanje u sklopu UNIC matematičkog seminara pod naslovom:
"Elliptic curves and Diophantine m-tuples".
Predavanje će biti održano s početkom u 16.00 sati putem platforme MS teams: Click here to join the meeting
Sažetak:
In this talk, we will describe some connections between Diophantine m-tuples and elliptic curves. A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus. It is known that there are infinitely many Diophantine quadruples in integers (the first example, the set {1,3,8,120}, was found by Fermat),and He, Togbe and Ziegler proved recently that there are no Diophantine quintuples in integers. Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. It is still an open question whether there exists any rational Diophantine septuple. We will describe several constructions of infinite families of rational Diophantine sextuples (this is joint work with M. Kazalicki, M. Mikic, V. Petricevic and M. Szikszai). These constructions use properties of corresponding elliptic curves. We will also show how Diophantine m-tuples can be used in the construction of high-rank elliptic curves over Q and Q(t) with given torsion group (this is joint work with J. Aguirre and J. C. Peral).
Pristupačnost
