MATHEMATICS SEMINAR by Benjamin Matschke

Time: 16:00
Location: SCI 103


Speaker               : Benjamin Matschke, Université de Bordeaux

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

Date                      : February 23, 2017 Thursday

Time                     : 16:00  

Cookie & Tea     : SCI 103 15:45

Place                    : SCI 103

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.