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Drinfeld’s ‘shtukas’ and the Ramanujan-Petersson conjecture. (Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson.) (French) Zbl 0899.11026
Astérisque. 243. Paris: Société Mathématique de France, 329 p. (1997).
The author extends a number of results of Drinfeld concerning the Langlands correspondence for the $$\text{GL}(2)$$-case to the higher rank situation. The main theorem he obtains in doing this is a large part of the Ramanujan-Petersson conjecture for the function field case. The main tool for proving such results are generalizations of Drinfeld’s concept of the so-called “shtukas” to the case of higher rank vector bundles by combining it with the concept of $${\mathcal D}$$-elliptic sheaves of the paper of G. Laumon, M. Rapoport and U. Stuhler [Invent. Math. 113, 217-338 (1993; Zbl 0809.11032)] by considering an additional action of a sheaf of division algebras, which makes “things more compact”, at least for the case $$r=1$$.
The first three chapters of the book contain a study of $${\mathcal D}$$-shtukas of arbitrary rank. Chapter I proves representability of the functor of rank $$r$$ $${\mathcal D}$$-shtukas with level structure as an algebraic stack $$\text{Cht}_{{\mathcal D},I}^r$$. Chapter II studies questions of reducibility of $${\mathcal D}$$-shtukas which occurs only for rank $$r\geq 2$$ and lead to some interesting modifications of the Harder-Narasimhan filtration of vector bundles on curves. Chapter III studies these concepts over finite fields and gives the adelic description of shtukas which ultimately comes down to the description of a vector bundle as a system of lattices. Chapter IV studies the special case of rank $$r=1$$, which is easier in several respects. Finally, in chapters V and VI, the calculation of Lefschetz numbers (traces of operators of type (Frobenius)$$\times$$(Hecke) on cohomology) and the evaluation of the Arthur-Selberg trace formula and its application to the Ramanujan-Petersson conjecture are done.

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11F70 Representation-theoretic methods; automorphic representations over local and global fields 14G25 Global ground fields in algebraic geometry 14L05 Formal groups, $$p$$-divisible groups 11S37 Langlands-Weil conjectures, nonabelian class field theory 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11R58 Arithmetic theory of algebraic function fields 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)