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 $b_{ c,d}=\lceil{ cd\over { c+d}}\rceil$. This proves Traverso's isogeny conjecture for $p$-divisible groups over $k$.
Categories: math.NT math.AG