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Traverso’s isogeny conjecture for \(p\)-divisible groups. (English) Zbl 1167.14322

Summary: Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(c, d\in\mathbb{N}\). Let \(b_{c,d}\geq 1\) be the smallest integer such that for any two \(p\)-divisible groups \(H\) and \(H'\) over \(k\) of codimension \(c\) and dimension \(d\) the following assertion holds: If \(H[p^{b_{c,d}}]\) and \(H'[p^{b_{c,d}}]\) are isomorphic, then \(H\) and \(H'\) are isogenous. We show that \(b_{c,d}=\lfloor\frac{cd}{c+d}\rfloor+1\). This proves Traverso’s isogeny conjecture for \(p\)-divisible groups over \(k\).

MSC:

14L05 Formal groups, \(p\)-divisible groups
11G10 Abelian varieties of dimension \(> 1\)
11G18 Arithmetic aspects of modular and Shimura varieties
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