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Breuil-Kisin modules and integral $p$-adic Hodge theory
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abstract
We construct a category of Breuil-Kisin $G_K$-modules to classify integral semi-stable Galois representations. Our theory uses Breuil-Kisin modules and Breuil-Kisin-Fargues modules with Galois actions, and can be regarded as the algebraic avatar of the integral $p$-adic cohomology theories of Bhatt-Morrow-Scholze and Bhatt-Scholze. As a key ingredient, we classify Galois representations that are of finite $E(u)$-height.
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Moduli stacks of \'etale (phi,Gamma)-modules and the existence of crystalline lifts
Every mod p representation of the absolute Galois group of a p-adic local field lifts to a crystalline representation of regular Hodge-Tate weights, via the geometry of new moduli stacks of etale (phi,Gamma)-modules.
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