## 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

#### U. N. Cordova, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakb06cbbcb8ee829802dc45288740da39f').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyb06cbbcb8ee829802dc45288740da39f = '&#97;p&#97;c&#101;tt&#105;' + '&#64;'; addyb06cbbcb8ee829802dc45288740da39f = addyb06cbbcb8ee829802dc45288740da39f + 'f&#97;m&#97;f' + '&#46;' + '&#117;nc' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r'; var addy_textb06cbbcb8ee829802dc45288740da39f = '&#97;p&#97;c&#101;tt&#105;' + '&#64;' + 'f&#97;m&#97;f' + '&#46;' + '&#117;nc' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r';document.getElementById('cloakb06cbbcb8ee829802dc45288740da39f').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyb06cbbcb8ee829802dc45288740da39f + '\'>'+addy_textb06cbbcb8ee829802dc45288740da39f+'<\/a>';

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.