Session S09 - Number Theory in the Americas

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

Hecke characters and some diophantine equations

Ariel Pacetti

U. N. Cordova, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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.

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