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Register a new userp-adic Monodromy of the Universal Deformation of a HW-cyclic Barsotti-Tate Group
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Authors
Yichao Tian
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Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the algebraic local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ over $S$, and by $Usubset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $rho$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $rho$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivalent to that Oorts $a$-number of $G_0$ equals 1, and it is satisfied by all connected one-dimensional Barsotti-Tate groups over $k$.
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Let $G$ be a connected reductive group over a $p$-adic local field $F$. We propose and study the notions of $G$-$varphi$-modules and $G$-$(varphi, abla)$-modules over the Robba ring, which are exact faithful $F$-linear tensor functors from the category of $G$-representations on finite-dimensional $F$-vector spaces to the categories of $varphi$-modules and $(varphi, abla)$-modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlayas slope filtration theorem in this context, and show that $G$-$(varphi, abla)$-modules over the Robba ring are $G$-quasi-unipotent, which is a generalization of the $p$-adic local monodromy theorem proven independently by Y. Andre, K. S. Kedlaya, and Z. Mebkhout.
Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that $D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over $k$ of the same codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers $gamma_D(i):=dim(pmb{Aut}(D[p^i]))$. We show that $n_Dle cd$, $(gamma_D(i+1)-gamma_D(i))_{iinBbb N}$ is a decreasing sequence in $Bbb N$, for $cd>0$ we have $gamma_D(1)<gamma_D(2)<...<gamma_D(n_D)$, and for $min{1,...,n_D-1}$ there exists an infinite set of truncated Barsotti--Tate groups of level $m+1$ which are pairwise non-isomorphic and lift $D[p^m]$. Different generalizations to $p$-divisible groups with a smooth integral group scheme in the crystalline context are also proved.
Let $S$ be the spectrum of a complete discrete valuation ring with fraction field of characteristic 0 and perfect residue field of characteristic $pgeq 3$. Let $G$ be a truncated Barsotti-Tate group of level 1 over $S$. If ``$G$ is not too supersingular, a condition that will be explicitly expressed in terms of the valuation of a certain determinant, we prove that we can canonically lift the kernel of the Frobenius endomorphism of its special fibre to a subgroup scheme of $G$, finite and flat over $S$. We call it the canonical subgroup of $G$.
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,din mathbb{N}$ be such that $h=c+d>0$. Let $H$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. For $minmathbb{N}^ast$ let $H[p^m]=ker([p^m]:Hrightarrow H)$. It is a finite commutative group scheme over $k$ of $p$ power order, called a Barsotti-Tate group of level $m$. We study a particular type of $p$-divisible groups $H_pi$, where $pi$ is a permutation on the set ${1,2,dots,h}$. Let $(M,varphi_pi)$ be the Dieudonne module of $H_pi$. Each $H_pi$ is uniquely determined by $H_pi[p]$ and by the fact that there exists a maximal torus $T$ of $GL_M$ whose Lie algebra is normalized by $varphi_pi$ in a natural way. Moreover, if $H$ is a $p$-divisible group of codimension $c$ and dimension $d$ over $k$, then $H[p]cong H_pi[p]$ for some permutation $pi$. We call these $H_pi$ canonical lifts of Barsotti-Tate groups of level $1$. We obtain new formulas of combinatorial nature for the dimension of $boldsymbol{Aut}(H_pi[p^m])$ and for the number of connected components of $boldsymbol{End}(H_pi[p^m])$.
Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.
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