Generalized Fermat equations over totally real fields

Wiles’ famous proof of Fermat’s Last Theorem pioneered the so-called modular method, in which modularity of elliptic curves is used to show that all integer solutions of the Fermat’s equation are trivial.

In this talk, we briefly sketch a variant of the modular method described by Freitas and Siksek in 2014, proving that for sufficiently large exponents, Fermat’s Last Theorem holds in five-sixths of real quadratic fields. We then extend this method to explore solutions to two broader Fermat-type families of equations. The main ingredients are modularity, level lowering, image of inertia comparisons, and S-unit equations.