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.