Mathematics – Number Theory
Scientific paper
2011-12-07
Mathematics
Number Theory
50 pages
Scientific paper
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$. 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 $n_D$, principles that for $m\in\Bbb N^{\ast}$ govern lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers $\gamma_D(i):=\dim(\pmb{Aut}(D[p^i]))$ which (like Newton polygons) go up under specializations. We show that $n_D\le cd$, that $(\gamma_D(i+1)-\gamma_D(i))_{i\in\Bbb N}$ is a decreasing sequence in $\Bbb N$, and that for $cd>0$ we have $\gamma_D(1)<\gamma_D(2)<...<\gamma_D(n_D)$. If $m\in\{1,...,n_D-1\}$, we also show that there exists an infinite set of truncated Barsotti--Tate groups of level $m+1$ over $k$ which are pairwise non-isomorphic and which lift $D[p^m]$. Different generalizations to $p$-divisible groups endowed with a smooth integral group scheme in the crystalline context are also proved.
Gabber Ofer
Vasiu Adrian
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