Local Shtukas, Hodge-Pink Structures and Galois Representations

Abstract

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.

56 pages, v3: final version published in "t-motives: Hodge structures, transcendence and other motivic aspects", Editors G. Böckle, D. Goss, U. Hartl, M. Papanikolas, European Mathematical Society Congress Reports 2020

References in corpus (2)

• Local Shtukas and Divisible Local Anderson Modules
• Phi-modules and coefficient spaces

Cited by in corpus (7)

• Local Shtukas and Divisible Local Anderson Modules
• Periods of Drinfeld modules and local shtukas with complex multiplication
• Hodge-Iwasawa Theory I
• Arithmetic Satake compactifications and algebraic Drinfeld modular forms
• The Carlitz Logarithm as a Period Morphism for Local $G$-Shtukas
• Product Formulas for Periods of CM Abelian Varieties and the Function Field Analog
• Category and Cohomology of Hodge-Iwasawa Modules