Seminario Teoría de Números

DIOPHANTINE APPLICATIONS OF SERRE’S MODULARITY CONJECTURE


SPEAKER: George Turcas

(University of Warwick, UK, and IMAR, Bucharest)

DATE & TIME: Tuesday, 24th September 2019 - 10:30

VENUE: Aula 420, Departamento de Matemáticas, UAM

ORGANISER: UAM - ICMAT

ABSTRACT: Successful resolutions of Diophantine equations over Q via Frey elliptic curves and modularity rest on three pillars: Mazur's isogeny theorem, modularity of elliptic curves defined over Q and Ribet's level-lowering theorem. One can replace the last two with Serre's modularity conjecture over Q, now a theorem due to Khare and Wintenberger. For general number fields K, there's no analogue of Mazur's isogeny theorem, but there is a formulation of Serre's modularity conjecture. In this talk, we will show how one can use the latter for showing that certain Diophantine equations do not have solutions in K.