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).
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/246