## 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 $K$.

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