# An Essay on the Prize in Mathematical Sciences 2007

The work of Robert Langlands and Richard Taylor, taken together, provides us with an extraordinary unifying vision of mathematics. This vision begins with "Reciprocity", the fundamental pillar of arithmetic of previous centuries, the legacy of Gauss and Hilbert. Langlands had the insight to imbed Reciprocity into a vast web of relationships previously unimagined. Langlands' framework has shaped – and will continue to shape, unify, and advance - some of the most important research programmes in the arithmetic of our time as well as the representation theory of our time. The work of Taylor has, by a route as successful as it is illuminating, established - in the recent past - various aspects of the Langlands programme that have profound implications for the solution of important open problems in number theory.

For a prime number p form the (seemingly elementary) function that associates to an integer n the value +1 if n is a square modulo p, the value – 1 if it isn't, and the value 0 if it is divisible by p. It was surely part of Langlands' initial vision that such functions and their number theory might be relatively faithful guides to the vast number-theoretic structure concealed in the panoply of automorphic forms associated to general algebraic groups. Langlands, viewing automorphic forms as certain kinds of representations (usually infinite-dimensional) of algebraic groups, discovered a unification of the two subjects, number theory and representation theory, that has provided mathematics with the astounding dictionary it now is in the process of developing and applying. Namely, the Langlands Philosophy: a dictionary between number theory and representation theory which has the uncanny feature that many elementary representation-theoretic relationships become - after translation by this dictionary – profound, and otherwise unguessed, relationships in number theory, and conversely.

In the mid 1960's Robert Langlands was one of the prime movers in the development of the general analytic theory of automorphic forms and their relationship to representation theory. Of particular note is his much celebrated general theory of Eisenstein series. Remarkably quickly after this, he was able to ennunciate in a rather precise way the audacious "Langlands philosophy" which has guided the subject ever since. This includes his extremely general "reciprocity conjecture" connecting automorphic forms with number theory and his "principle of functoriality", a beautiful conjecture that subsumes all these ideas in terms of internal properties of representations. In the 1970's and 1980's Langlands went on to attack many important special cases of his conjectures using generalizations of the Selberg Trace Formula. Of particular note is his theory of cyclic base change for GL(2), an example of "functoriality" which has profound applications to number theory. He pioneered the use of the trace formula to study Shimura varieties. He also laid out a very detailed blueprint (the theory of "endoscopy") on how to overcome deep problems that were encountered when trying to apply the trace formula to analyse Shimura varieties or to prove cases of functoriality. In sum, Langlands' insight offers us a grand unification, already used to establish some of the deepest advances in number theory in recent years.

Indeed, it is thanks to the work of Richard Taylor that we now have some of these advances. To cite the most recent of these breakthroughs, he and co-workers (Michael Harris, Laurent Clozel, and Nicholas Shepherd-Barron) have established an important part of a basic conjecture that has been around for 40 years. At the same time, they have extended - in a striking way - our ability to make use of Langlands' ideas, in combination with work of others, for arithmetic purposes. The technical statement of what they have done is to have proved the Sato-Tate conjecture for elliptic curves over totally real fields, provided that the curve has a place of multiplicative reduction. The Sato-Tate conjecture predicts that certain error terms in a broad class of important numerical functions of prime numbers conform to a specific probability distribution. In this recent work we see otherwise separate mathematical sub-disciplines coming together and connecting with each other in an illuminating way. Moreover, the successful strategy adopted, in keeping with Langlands' principle of functoriality, involves an infinite sequence of automorphic forms attached to algebraic groups of higher and higher rank. All this is surely just the beginning of a much bigger story, as envisaged by Langlands.

Richard Taylor's earlier work includes his celebrated collaboration with Wiles on the resolution of Fermat's Last Theorem followed by his quite significant contribution to the collaborative effort to finish fully the modularity of elliptic curves over the rational number field, his collaboration with Michael Harris culminating in the resolution of the local Langlands' Conjecture for the general linear group in n dimensions, and his work resolving the classical Artin conjecture for a quite important class of non-solvable Galois representations of degree two.

The work of Robert Langlands and Richard Taylor demonstrates the profundity and the vigour of modern number theory and representation theory. Together they amply deserve the honour of the Shaw Prize.

Mathematical Sciences Selection Committee

The Shaw Prize

11 September 2007, Hong Kong