# Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the restriction functor from $S$ to $U$ is an equivalence of categories. If this is the case, we say that the fibered category is pure (terminology coming initially from Zariski-Nagata's theorem on the purity of the ramification locus). The case of abelian schemes is particularly interesting.

Suppose first that all points of $S$ are of residual characteristic $0$. Then Grothendieck proved in "Un théorème sur les homomorphismes des schémas abéliens" (Inventiones mathematicae 1966/67, Volume 2, Issue 1, pp 59-78) that the category of abelian schemes satisfies purity (Corollaire 4.5). He then went on to ask (in paragraph 4.8) whether it is possible that this result remains true up to $p$-isogeny in characteristic $p$ (the $p$-isogeny is necessary, as I will explain shortly).

My question is the following: has there been progress on this question since ? More generally, in the general situation above, if $A$ is an abelian scheme over $U$, does there exists an abelian scheme $B$ over $S$ and an isogeny $B_U\rightarrow A$ over $U$ (ideally of degree divisible only by residual characteristics of $S$) ?.

State of the literature: the problem of extending $A$ to $S$ "on the nose" has been intensely studied. In the same paper (section 4.6), Grothedieck provides a simple example of an abelian scheme over $\mathbb{A}^2_k\setminus \{0\}$ for $k$ algebraically closed of positive characteristic not extending to $\mathbb{A}^2_k$. Later, Gabber-Raynaud-Oort provided similar but more sophisticated examples over some regular local rings of mixed characteristic of dimension $2$ (described in De Jong-Oort, "On extending families of curves", Journal of Algebraic Geometry, 6 (1997), pp. 545--562, paragraph 6). edit: the counter-examples can chosen to be principally polarized.

In mixed characteristic local rings with low ramification, there is nevertheless a positive result due to Vasiu-Zink (Theorem 3, Corollary 5 in "Purity results for p-divisible groups and abelian schemes over regular bases of mixed characteristic", preprint). The same paper constructs many new counter-examples (Lemma 27, Theorem 28).

As far as I can see, the counter-examples above are isogeneous to abelian schemes that do extend, so they do not answer my question.

Let me mention for completeness that the similar problem for curves has a much more satisfactory answer: purity is true for smooth projective $n$-pointed curves of genus $g$ when $2-2g-n<0$ (Moret-Bailly, C. R. Acad. Sci. Paris, 300 n° 14 (1985), 489–492.), for stable curves of constant topological type (the De Jong-Oort paper above). Even better, families of smooth projective curves with $2-2g-n<0$ are controlled by the étale fundamental group on normal base schemes (Stix, "A monodromy criterion for extending curves", Int Math Res Notices (2005) 2005 (29): 1787-1802. ).