## STNB2022 (35th edition)

### Hecke characters and $\mathbb{Q}$-curves in the modular method

#### Presenters

Lucas Villagra Torcomian

#### Abstract

In this talk we will see how we can twist Galois representations coming from Frey $\mathbb{Q}$-curves defined over quadratic fields to extend to the absolute Galois group of $\mathbb{Q}$, and then follow the modular method. More concretely, we will focus on solutions of equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$. This is based on works in collaboration with Ariel Pacetti.

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