Traverso's isogeny conjecture for p-divisible groups

Abstract

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

8 pages, laTex; to appear in Rend. Sem. Mat. Univ. Padova