Commentary on “an Elementary Introduction to the Langlands Program” by Stephen Gelbart

When I was asked to suggest a Bulletin article worthy of reprinting, I knew the answer right away: An Elementary Introduction to the Langlands Program by Steven Gelbart, published by the Bulletin in 1984, remains one of the best reviews of the Langlands Program, even though this subject has expanded tremendously in the intervening years. This paper has greatly influenced my own research and is in fact one of my all-time favorites. I still keep my old xerox copy on roughly cut cheap yellow paper, made in a Moscow library in my student years, as a memento. It is precisely the kind of review article that the Bulletin strives to publish: describing a fascinating area of mathematics in a way that is accessible to nonspecialists. Gelbart introduces in it the key concepts of the Langlands Program appealing to very little mathematical background. He starts with a brief recollection of basic number theory and a review of some “classical themes,” such as the Local-Global Principle in number theory, modular forms, and Artin’s L-functions. He then moves on to the key concept of automorphic representations of reductive groups over the adèles. This sets the stage for the conjectural Langlands’ correspondence relating n-dimensional representations of the Galois group of a number field (or a function field) F and automorphic representations of the group GLn over the ring of adèles of F . Introducing the Langlands dual group, Gelbart then presents Langlands’ general functoriality principle (which Langlands himself considers as the central tenet of his Program). The last chapter reviews what was known about all this at the time when Gelbart’s article went to press. So much has happened since then! Though it was conceived initially [12] as a bridge between number theory and harmonic analysis, the Langlands Program has moved to other areas of mathematics, such as geometry, and even to quantum physics. It is tempting to think of it as a “grand unified theory” of mathematics, since it ties together so many different disciplines.