Cohomology of arithmetic groups and Fermat's Last Theorem

We use two fundamental reciprocity conjectures in the Langlands Program that involve the cohomology of arithmetic groups and derive an algorithmically testable criterion for a number field K which, if satisfied, implies the truth of asymptotic Fermat's Last Theorem over K. Most imaginary quadratic fields satisfy the criterion. This is joint work with Samir Siksek (Warwick).