## Barcelona Fall Workshop on Number Theory II

### Diophantine applications of Serre's modularity conjecture

#### Presenters

George Turcas (Bucharest)

#### Abstract

Successful resolutions of Diophantine equations over $\mathbb{Q}$
via Frey elliptic curves and modularity rest on three pillars:
Mazur's isogeny theorem, modularity of elliptic curves defined over
$\mathbb{Q}$ and Ribet's level-lowering theorem. One can replace the
last two with Serre's modularity conjecture over $\mathbb{Q}$, 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 $K$.

#### Files

Download presentation.