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Minimal \(p\)-divisible groups. (English) Zbl 1081.14065
Let \(X_1\) and \(X_2\) be \(p\)-divisible groups. This paper gives a necessary and sufficient condition for \(X_1 [p] \simeq X_2 [p]\) to imply \(X_1 \simeq X_2\). For any pair \((m,n)\) of coprime nonnegative integers there is a \(p\)-divisible group \(H_{m,n}\) over \({\mathbb F}_p\) of dimension \(m\) with Serre dual of dimension \(n\) which is isosimple and with endomorphism ring \(\text{End}(H_{m,n} \otimes \overline{\mathbb F}_p)\) the maximal order in \(\text{End}^0 (H_{m,n} \otimes \overline{\mathbb F}_p)\) [see section 5 of A. J. de Jong, F. Oort, J. Am. Math. Soc. 13 No. 1 209–241 (2000; Zbl 0954.14007)]. If \(\beta\) is a Newton polygon then by associating to a segment of width \(m+n\) and slope \(n/m+n\) the pair \((m,n)\), \(\beta\) corresponds to a finite sum \(\Sigma_{i}(m_i, n_i)\) and so to a \(p\)-divisible group \(H(\beta) = \times_i H_{m_i, n_i}\). A \(p\)-divisible group \(x\) is said to be minimal if there is a Newton polygon \(\beta\) and isomorphism \(x_k \simeq H(\beta)_k\) for \(k\) an algebraically closed field. A \(1\)-truncated Barsotti-Tate group, \(G\), is minimal if \(G_k \simeq H(\beta) [p]_k\). The main theorem of this paper is that if \(X_1[p] \simeq G \simeq X_2[p]\) with \(G\) minimal, then \(X_1 \simeq X_2\) and if \(G\) is not minimal then there are infinitely many mutually nonisomorphic \(p\)-divisible groups \(X\) with \(X[p] \simeq G\).

MSC:
14L05 Formal groups, \(p\)-divisible groups
Citations:
Zbl 0954.14007
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