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Compactification of the stacks of shtukas and geometric invariant theory. (Compactification des champs de chtoucas et théorie géométrique des invariants.) (French) Zbl 1144.11047
Astérisque 313. Paris: Société Mathématique de France (ISBN 978-2-85629-243-3/pbk). 124 p. (2007).
This book contains the results announced previously by the author in [C. R., Math., Acad. Sci. Paris 340, No. 2, 147–150 (2005; Zbl 1082.11036)]. A crucial step in the proof of the Langlands correspondence for $$\text{GL}_r$$ over function fields is the construction and study of a compactification of the stack of rank $$r$$ shtukas (by Drinfeld for $$r=2$$ and Lafforgue for arbitrary $$r$$).
In this text the author uses Geometric Invariant Theory (à la I. V. Dolgachev and Y. Hu [Publ. Math., Inst. Hautes Étud. Sci. 87, 5–56 (1998; Zbl 1001.14018)] and M. Thaddeus [J. Am. Math. Soc. 9, No. 3, 691–723 (1996; Zbl 0874.14042)]) to construct compactifications of the stack of rank $$r$$ shtukas. This novel approach does not only recover Lafforgue’s original compactification, but also produces several new ones.
It is hoped that this method will be more suitable to attack the problem of compactifying the stacks of $$G$$-shtukas for arbitrary reductive groups $$G$$.

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11R39 Langlands-Weil conjectures, nonabelian class field theory 14D20 Algebraic moduli problems, moduli of vector bundles 11-02 Research exposition (monographs, survey articles) pertaining to number theory