# Robert Langlands describes James Arthur's research

In Robert Langlands' nomination of James Arthur for President of the American Mathematical Society he gives a brief overview of Arthur's research contributions up to 2003. We present a version of the brief description of Arthur's work from the nomination, see R P Langlands, Nomination for President Elect: James G Arthur,

*Notices of the American Mathematical Society***50**(10) (2003), 1303-1305.**James Arthur's research contributions.**

Jim's name is attached to the formula or technique in the theory of automorphic forms that is referred to either simply as the trace formula or, frequently, as the Arthur- Selberg trace formula. He has devoted the bulk of his mathematical efforts to it. On the website www.sunsite.ubc.ca/DigitalMathArchive/Langlands the interested reader can find a short, historically oriented general introduction to automorphic forms, the trace formula, and Arthur's work that was written as a supplement to the all too brief sketch that follows, as well as a much longer appreciation that appeared in the

*Canad. Math. Bull.*

**44**(2001), 160-209 on the occasion of the award of the Canada Gold Medal and that attempts a survey less of the area as a whole than of Arthur's many contributions to it. Here I shall say no more than is necessary to underline the scope and difficulty of his work and its great importance for number theory at the present time and in the future.

The role of the theory of automorphic forms in modern number theory is more familiar than it once was because of the famous Taniyama-Shimura-Weil conjecture, its proof by Wiles, and its application to the Fermat theorem. The deep questions with which we are confronted when attempting to classify and understand algebraic irrationalities and the strikingly beautiful answers to them suggested by the theory are none the less hardly as familiar as they might be. The origins of the modern theory of automorphic forms lie to a considerable degree in the extension not only of quadratic reciprocity to the higher reciprocity laws but also of Gauss's analysis of the arithmetic of roots of unity, thus of the construction of regular polygons, to more recondite irrationalities, those associated to the division of elliptic curves. However, it has other roots as well - quite different - in analysis, both real and complex, in geometry, and in representation theory. It is the sophistication of aims together with the sophistication of proposed techniques that render the subject difficult.

The trace formula itself is an analytic technique that is used to investigate the spectral theory of the homogeneous spaces that link the analysis and the number theory. It is not difficult analytically if the homogeneous space is compact but still very important as Selberg discovered. When the space is not compact but of rank one, the formula is not only important but also difficult and is due to Selberg. For groups of higher rank, where the analytic difficulties are much more severe, it is the work of Arthur.

At the core of the formula in higher rank is the simultaneous spectral theory of several commuting differential operators, whereas in rank one there is only a single operator. The problems to be solved, first of all to obtain a formula in higher rank and then to turn it into an effective tool, lie in many domains: Fourier transforms in several variables, ordinary differential equations, measure theory, convex bodies, local harmonic analysis on real and on $p$-adic groups.

Arthur has not only had to develop a variety of techniques to handle them but has been led to some deep and important conjectures in representation theory - one global, related to the Ramanujan conjecture, and one local, related to the classification of unitary representations of reductive groups over a local field. Both these conjectures have had a great influence on the work of a number of important mathematicians such as Vogan, Waldspurger, and Moeglin.

Although much work and many deep discoveries remain before the full arithmetic depth of the trace formula reveals itself, Arthur has already explored profoundly many aspects of the trace formula, especially invariance and stabilisation, and in part in collaboration with Clozel, has made a number of important applications to the transfer of automorphic forms from one form of the general linear group to another or from symplectic and orthogonal groups to a general linear group. His papers will, I believe, be essential reading for those in the field for a long time to come.

Others, too, have found striking applications of the trace formula to major arithmetical conjectures. Kottwitz's proof of an important conjecture of Weil on the volumes of arithmetic quotients exploited the first forms of Arthur's trace formula. The formula of Arthur, but over function fields and not number fields, was an essential tool in the work for which Lafforgue was recently awarded the Fields Medal.

Last Updated March 2024