Koç University, Mathematics Seminar

Date & Time: Thursday, February 23, 16:00-17:00

Place: SCI 103

Speaker:  Benjamin Matschke,  Université de Bordeaux
 

Title: Solving several classical Diophantine equations via the Shimura-Taniyama conjecture 

Abstract: In this talk we present a project in which we constructed practical algorithms to solve S-unit, Mordell, cubic Thue, cubic Thue--Mahler, as well as generalized Ramanujan--Nagell equations, and to compute S-integral points on rational elliptic curves with given Mordell--Weil basis. Our algorithms rely on new height bounds, which we obtained using the method of Faltings (Arakelov, Parshin, Szpiro) combined with the Shimura--Taniyama conjecture (without relying on linear forms in logarithms), as well as several improved and new sieves. As one application we obtained a table of all rational elliptic curves with good reduction outside certain finite sets of primes, including the set {2, 3, 5, 7, 11}, and all sets whose product is at most 1000.
In addition we used the resulting data to motivate several conjectures and questions, such as Baker's explicit abc-conjecture, and a new conjecture on the number of S-integral points of rational elliptic curves. This is joint work with Rafael von Känel.