Benjamin Matschke (MPIM)

22 October 2015 - W01 0-012 (Wechloy), 4 pm c.t.

On solving S-unit and Mordell equations using the Shimura-Taniyama conjecture

Abstract: With Rafael von Känel we construct practical algorithms for solving S-unit and Mordell equations. They are based on explicit height bounds as well as direct enumeration algorithms that improve many previous works. To demonstrate the usefulness of these algorithms, we solve large classes of S-unit and Mordell equations as well as some Thue and Thue-Mahler equations. The data obtained motivate some questions (e.g., it can be used to test Baker's explicit abc conjecture). Our methods are based on the Shimura-Taniyama conjecture as well as on methods of Faltings (Arakelov, Parshin, Szpiro), and they do not require the theory of logarithmic forms.