## 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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### References:

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