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Let $k$ be an algebraically closed field of positive characteristic $p$. We first classify the $D$-truncations mod $p$ of Shimura $F$-crystals over $k$ and then we study stratifications defined by inner isomorphism classes of these $D$-truncations. This generalizes previous works of Kraft, Ekedahl, Oort, Moonen, and Wedhorn. As a main tool we introduce and study Bruhat $F$-decompositions; they generalize the combined form of Steinberg theorem and of classical Bruhat decompositions for reductive groups over $k$.
Let $F$ be a non-archimedean local field with residue field $mathbb{F}_q$ and let $G = GL_2/F$. Let $mathbf{q}$ be an indeterminate and let $H^{(1)}(mathbf{q})$ be Vigneras generic pro-p Iwahori-Hecke algebra of the p-adic group $G(F)$. Let $V_{wideh
We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and other endo
We study the mod $p$-points of the Kisin-Pappas integral models of abelian type Shimura varieties with parahoric level structure. We show that if the group is quasi-split and unramified, then the mod $p$ isogeny classes are of the form predicted by t
Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $mathbb{F}_p$. We show that the supersingular irreducible $mathcal{H}^{(1)}$-modules of dim
This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $mathbf{GL_2}(F)$. In arXiv:2007.01364v1 we have co