An appreciation

Qing Zou asked me to contribute some remarks introductory to the book that follows. I have not read it. I have indeed only seen a relatively small number of pages. Moreover, although in the course of my career as a mathematician and, in general, in the course of my life, I have acquired a working knowledge of several languages, Chinese in any of its forms is not one of them. Consequently I have not read even the pages received and can only guess at their content and that of the text as a whole. So this appreciation consists of jottings suggested by my own reflections over the years, but not entirely unrelated to the pages at hand. Since the phrase Langlands programme makes me uneasy, I would like to think of the appreciation as a collection of partly autobiographical but largely historical comments on those parts of pure mathematics, of which there are surprisingly many, that appear in the contemporary theory of automorphic forms or automorphic representations. They have been embraced by it — or coupled to it — over the course of the nineteenth, twentieth, and now of the twenty-first centuries with, I believe, an inevitability that is hard to deny. My comments here are not unrelated to those in the lecture Is there beauty in mathematical theories? (see The many faces of beauty, ed. Vittorio Hösle) but are perhaps more mature. Nevertheless, they still remain in good part provisional. In contrast to those earlier remarks, what follows is directed to individuals with an active interest in mathematics, in particular to the readers of Qing Zou’s book. So I do not hesitate to suggest topics on which they might reflect. The current theory of automorphic representations, has a number of aspects that it is important to distinguish from each other, either because one is at a higher conceptual level than another or because — their resemblance, which is often compelling, notwithstanding — they refer to mathematical domains with different techniques and different histories. On some I have reflected for long periods, on others for short periods, and on some of the most important not at all. Their classical manifestations appeared with: (i) the quadratic reciprocity law and its development as class-field theory in the hands of Weber, Takagi, Artin and others; (ii) with Cartesian geometry either in the form of algebraic geometry or of differential geometry, where Lefschetz, Hodge or Chern are familiar names, not to speak of Riemann or the discoverers of non-Euclidean geometry; (iii) with the theory of groups and their representations, to which are attached, among others, the names of Galois, Frobenius, and in recent times Harish-Chandra; (iv) algebraic geometry over finite fields, in which two representative names are Weil and Grothendieck; (v) the developments arising from the classical theory of elliptic modular functions, especially p-adic deformations, whose importance Serre has emphasized and which feature prominently in the proof of Fermat’s theorem byWiles, and the theory of Shimura varieties; (vi) the theory of automorphic forms on general groups as developed, even rescued, by Siegel largely as a continuation of the work on quadratic forms, by a large number of prominent nineteenth-century mathematicians, Eisenstein, Minkowski and others — this is not the place for a careful history and I am not the one to prepare it — and greatly enriched by Maaß and then Selberg and others as the analytic theory of automorphic forms. There are other currents appearing simultaneously