Dimensions of automorphism group schemes of finite level truncations of 𝐹-cyclic 𝐹-crystals

Ding, Zeyu; Xiao, Xiao

doi:10.1090/tran/8243

Dimensions of automorphism group schemes of finite level truncations of $F$-cyclic $F$-crystals
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by Zeyu Ding and Xiao Xiao PDF

Trans. Amer. Math. Soc. 374 (2021), 269-302 Request permission

Abstract:

Let $\mathcal {M}_{\pi }$ be an $F$-cyclic $F$-crystal over an algebraically closed field of positive characteristic defined by a permutation $\pi$ and a set of prescribed Hodge slopes. We prove combinatorial formulas for the dimension $\gamma _{\mathcal {M}_{\pi }}(m)$ of the automorphism group scheme of $\mathcal {M}_{\pi }$ at finite level $m$ and the number of connected components of the endomorphism group scheme at finite level $m$. As an application, we show that if $\mathcal {M}_{\pi }$ is a nonordinary Dieudonné module defined by a cycle $\pi$, then $\gamma _{\mathcal {M}_{\pi }}(m+1) - \gamma _{\mathcal {M}_{\pi }}(m) < \gamma _{\mathcal {M}_{\pi }}(m) - \gamma _{\mathcal {M}_{\pi }}(m-1)$ for all $1 \leq m \leq n_{\mathcal {M}_{\pi }}$, where $n_{\mathcal {M}_{\pi }}$ is the isomorphism number of $\mathcal {M}_{\pi }$.

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