Reciprocity in the Langlands program since Fermat’s Last Theorem

Abstract

The reciprocity conjecture in the Langlands program links motives to automorphic forms. The proof of Fermat’s Last Theorem by Wiles in (Taylor and Wiles 1995) and (Wiles 1995) introduced new tools to study reciprocity. This survey reports on developments using these ideas (and their generalizations) in the last three decades.