# zbMATH — the first resource for mathematics

Local shtukas, Hodge-Pink structures and Galois representations. (English) Zbl 1440.14113
Böckle, Gebhard (ed.) et al., $$t$$-motives: Hodge structures, transcendence and other motivic aspects. EMS Series of Congress Reports. Zürich: European Mathematical Society (EMS). 183-259 (2020).
Summary: We review the analog of Fontaine’s theory of crystalline $$p$$-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois representations are replaced by the Tate modules of so-called local shtukas. We prove that the Tate module functor is fully faithful. In addition to this étale realization of a local shtuka we discuss also the de Rham and the crystalline cohomology realizations and construct comparison isomorphisms between these realizations. We explain how local shtukas and these cohomology realizations arise from Drinfeld modules and Anderson’s $$t$$-motives. As an application we construct equi-characteristic crystalline deformation rings, establish their rigid-analytic smoothness and compute their dimension.
For the entire collection see [Zbl 1441.14003].

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 11F70 Representation-theoretic methods; automorphic representations over local and global fields 14G20 Local ground fields in algebraic geometry 14G35 Modular and Shimura varieties 11G09 Drinfel’d modules; higher-dimensional motives, etc.
Full Text: