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Seminario Teoría de Números UAM-ICMAT

Seminario Teoría de Números UAM-ICMAT

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.

Location DATE & TIME: Tuesday, 24th September 2019 - 10:30 VENUE: Aula 420, Departamento de Matemáticas, UAM