## View abstract

### Session S09 - Number Theory in the Americas

Thursday, July 22, 16:30 ~ 17:00 UTC-3

## Hecke characters and some diophantine equations

### Ariel Pacetti

In this talk we will study solutions to the equation $x^2 + dy^6 = z^p$. We will explain how to attach a ${\mathbb Q}$-curve over a quadratic field to a putative solution, and how to extend the representation to a rational one of a concrete level and Nebentypus (corresponding to a classical modular form via Serre's conjectures). The way to explicitly obtain the level and nebentypus is via the construction of a Hecke character with some desired properties. After computing the respective space of classical modular forms, some classical elimination techniques allow us to prove that no non-trivial solution exists for some values of $d$ and all $p$ sufficiently large.