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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
##### Keywords:
Barsotti-Tate group; Newton polygon
Zbl 0954.14007
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