Analogues of Traverso's Isogeny Conjecture for Some Unitary p-Divisible Groups

Abstract

Let $p$ be a prime and $\mathbb{k}$ an algebraically closed field of characteristic $p$. Two invariants worthy of study for a $p$-divisible group $X$ over $\mathbb{k}$ are its isogeny cutoff $b_X$, which measures the amount of $p$-torsion necessary to determine the Newton polygon of $X$, and its minimal height $q_X$, which measures its distance to the closest minimal $p$-divisible group (in the sense of Oort). Traverso's Isogeny Conjecture predicts a bound on $b_X$ in terms of its height and dimension. The conjecture was proved in [NV07] and a refinement was given in [LNV13]; the latter also gives a bound on $q_X$. In this paper, we study these invariants when restricting to the class of supersingular unitary $p$-divisible groups $X$ of signature $(a,g-a)$, where $0<a<g$, for $p>2$, and prove further refinements on their bounds purely in terms of $a$. In particular, we provide a complete description of the possible values of $q_X$. Finally, we rephrase our results in the language of the ${\mathrm{BT}}_m$ stratifications of characteristic-$p$ fibers of Shimura varieties for $\mathrm{GU}(a,g-a)$ in the case of an inert prime $p>2$.

arXiv commentNew title and abstract. Revised introduction and exposition, including proofs of some claims for completeness (Lemma 3.2 and Lemma 4.4); minor errors corrected. A new construction (Example 5.6) yields a strengthening of Theorem 5.6 of v1 (now Theorem 5.8). Statement and proof of Corollary 6.2 of v1 (now Corollary 6.3) clarified