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We review the analog of Fontaine's theory of crystalline $p$-adic Galoisrepresentations and their classification by weakly admissible filteredisocrystals in the arithmetic of function fields over a finite field. Therecrystalline Galois representations are replaced by the Tate modules ofso-called local shtukas. We prove that the Tate module functor is fullyfaithful. In addition to this \'etale realization of a local shtuka we discussalso the de Rham and the crystalline cohomology realizations and constructcomparison isomorphisms between these realizations. We explain how localshtukas and these cohomology realizations arise from Drinfeld modules andAnderson's $t$-motives. As an application we construct equi-characteristiccrystalline deformation rings, establish their rigid-analytic smoothness andcompute their dimension.

Local Shtukas, Hodge-Pink Structures and Galois Representations