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.

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.

Reviewer: U.Stuhler (Göttingen)

##### 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) |