The trace formula and its applications: An introduction to the work of James Arthur (Q2732640)

This is a remarkable paper in which a mathematician whose ideas have determined the direction of the development of the theory of automorphic forms perhaps more than any other pays generous tribute to his one-time student J. Arthur on the occasion of the latter's being awarded the Canada Gold Medal. NEWLINENEWLINENEWLINEThe main part of the paper is taken up with a discussion of Arthur's work on the trace formula. It has become clear that for a large number of questions about modular forms and their generalizations that of comparison between the set of automorphic (or admissible) representations associated with different groups and global (or local) fields is important. The most natural method to approach such problems is the trace formula. In fact, this has been clear for a long time now, at least since \textit{M. Eichler}'s first paper on the trace formula [J. Indian Math. Soc., New Ser. 20, 163-206 (1956; Zbl 0073.26501)], presented at the same conference at which Selberg presented his version, where, in effect using the language of Brandt matrices and theta functions, Eichler compares different orthogonal groups with one another. However Langlands has stressed many time the central importance of the systematic use of the trace formula (coupled with the Gelfand-Kazhdan) in the general context as the proper way to understand the totality of automorphic representations. Although the trace formula is very natural, it turns out that it is also technically very intricate, and is correspondingly difficult to apply. J. Arthur has devoted an enormous amount of effort to the problems that arise, but there is no synoptic account available for the mathematician who might have need of it. In this article Langlands shares many of his insights with the rest of us. As he himself makes clear this, is by no means an exhaustive treatment but rather a series of reflections on a number of central points, which will be invaluable to the serious scholar. NEWLINENEWLINENEWLINEThe reviewer, as a Northern Irish number theorist, feels compelled to take issue with two points. One is the very cabbalistic interpretation of Yeats' poem ``An Irish airman foresees his death''; the other the somewhat tendentious remarks in the introduction about number theorists which do not take account of the breadth of number theory in its entirety, nor its inherent organic disordliness. Representation theoretic methods have a place, an important place, in this garden, but there is much else there besides.