Canonical subgroups of p-divisible groups with $\mathcal{O}_F$-action: existence beyond the current residue-characteristic and Hasse-invariant bounds, and partial degrees without Hypothesis 0.1

For a p-divisible group $G$ over the ring of integers $\mathcal{O}_K$ of a complete valued extension of $\mathbb{Q}_p$, equipped with an action of the ring of integers $\mathcal{O}_F$ of a finite (possibly totally ramified) extension $F/\mathbb{Q}_p$, the theory of the canonical subgroup predicts a distinguished finite flat subgroup $C\subset G[p]$ lifting the kernel of the Frobenius on the special fiber whenever $G$ is close to ordinary. Existence is currently known only under restrictive hypotheses: Fargues's theorem gives it when the Hasse invariant satisfies $ha(G)<1/2$ with $p>3$ (or $ha(G)<1/3$ for $p=3$), and the refined partial-degree machinery of Bijakowski, which computes the partial degrees of $C$, of its $\pi$-power pieces $C[\pi^k]$ (where $\pi$ denotes a uniformizer of $\mathcal{O}_F$, a generator of its maximal ideal), and of the successive quotients $G/C[\pi^k]$ from the partial Hasse invariants, is proved only under Hypothesis 0.1 (freeness of $\omega_G\otimes K$ over $K\otimes\mathcal{O}_F$) and smallness bounds such as $ha(G)<1/e$ and $ha(G)<1/(pe)$. The problem is to construct the canonical subgroup, its $\pi$-power filtration pieces, and the canonical subgroups of the successive quotients under hypotheses as weak as possible on the Hasse invariant and without any restriction on the residue characteristic $p$ (including $p=2$), and to extend the partial-degree and partial-Hasse-invariant theory beyond Hypothesis 0.1 to general signatures, respecting the necessary constraints that, in the source's normalization, existence of a canonical subgroup already forces $ha(G)<1/2$, and that (by the source's Remark 3.7) failure of Hypothesis 0.1 forces $ha(G)=1$, so at general signatures the target is the refined invariant theory rather than the existence of the classical canonical subgroup.

Background

A $p$-divisible group (or Barsotti–Tate group) $G$ over a base ring is an inductive system of finite flat group schemes $G[p^n]$ of $p^n$-torsion, each killed by $p^n$, with $G[p^{n+1}]/G[p^n]\simeq G[p]$; abelian varieties and formal groups give the main examples. Writing $\omega_G$ for the module of invariant differentials of $G$ along its unit section, the Verschiebung map $V:\omega_G\to\omega_{G^{(p)}}$ (the dual of relative Frobenius, with the superscript $(p)$ denoting a twist by the Frobenius on the base) has a determinant; its truncated $p$-adic valuation $ha(G)\in[0,1]$, normalized by $v(p)=1$, is the Hasse invariant. It vanishes exactly when $G$ is ordinary, i.e. when $G[p]$ splits into a multiplicative part and an étale part.

When an elliptic curve or, more generally, a $p$-divisible group is close to ordinary ($ha(G)$ small), the kernel of the Frobenius on the special fiber lifts to a finite flat subgroup $C\subset G[p]$, called the canonical subgroup, which coincides with the multiplicative part of $G[p]$ in the ordinary limit. This was constructed by Lubin for formal groups, by Katz for elliptic curves, by Abbes–Mokrane and Tian for general Barsotti–Tate groups, and by Fargues by a purely local method. A key tool in the modern theory is Fargues's degree $\deg H\in[0,\mathrm{ht}(H)]$ of a finite flat group scheme $H$ over $\mathcal{O}_K$, defined as the valuation of the Fitting ideal of $\omega_H$; it behaves like a height-weighted measure of how multiplicative $H$ is. Fargues proved (for $p\neq 2$) that if $ha(G)<1/2$ (with the stricter bound $1/3$ for $p=3$), then there is a canonical subgroup $C\subset G[p]$ of height $\dim G$ with $\deg C^D=ha(G)$, where $C^D$ is the Cartier dual of $C$ (the group scheme $\mathrm{Hom}(C,\mathbb{G}_m)$ of characters); the same bound with $ha(G)<1/(2p)$ (resp. $1/(3p)$ for $p=3$) produces the canonical subgroup of the quotient $G/C$. For $p=2$, existence is known in the endomorphism-free setting through Hattori's approach via Breuil–Kisin modules (the classification theory that encodes a $p$-divisible group over $\mathcal{O}_K$ by a finite module over a power-series ring equipped with a Frobenius-semilinear map, turning statements about $G[p^n]$ into explicit linear algebra).

The source paper (Bijakowski, Canad. J. Math. 2018) refines this picture when $G$ carries an action of $\mathcal{O}_F$, the ring of integers of a finite extension $F/\mathbb{Q}_p$ with ramification index $e$ and residue degree $f$, and with uniformizer $\pi$ (a generator of the maximal ideal of $\mathcal{O}_F$), a structure arising from real multiplication on Hilbert modular varieties and, more generally, from PEL Shimura varieties. The module $\omega_G$ then decomposes according to the embeddings of $F$ into $\overline{\mathbb{Q}}_p$, and one can define partial Hasse invariants $ha^{[j]}_i(G)\in[0,1]$ (introduced in the unramified setting and for the ramified Hilbert case by Reduzzi–Xiao) as truncated valuations of determinants of the Verschiebung on the graded pieces of a filtration of $\omega_G$, and partial degrees $\deg^{[j]}_i H$ of an $\mathcal{O}_F$-stable finite flat subgroup $H\subset G[p]$ as truncated valuations of determinants of the maps induced on these graded pieces. These invariants are the building blocks for computing the refined structure of the canonical subgroup with its induced $\mathcal{O}_F$-action: the paper proves, under Hypothesis 0.1 — the freeness of $\omega_G\otimes_{\mathcal{O}_K}K$ over $K\otimes_{\mathbb{Z}_p}\mathcal{O}_F$, i.e. the condition that the dimensions of the eigenspaces of $\omega_G\otimes K$ be constant across the embeddings of $F$ (a strong form of 'no obstruction to being ordinary'; freeness of $\omega_G$ itself over $\mathcal{O}_K\otimes\mathcal{O}_F$ is the Rapoport condition) — that the partial degrees of the canonical subgroup and of its pieces $C[\pi^k]/C[\pi^{k-1}]$, and the partial Hasse invariants of the quotients $G/C[\pi^k]$, are given by explicit formulas in the partial and primitive Hasse invariants, provided $ha(G)<\min(1/e,1/2)$ and, for the quotient statements, $ha(G)<1/(pe)$ together with existence of a canonical subgroup for $G/C$. The paper also proves (Theorem 3.2) a converse of independent interest: any finite flat $C\subset G[p]$ of height $\dim G$ with $\deg C^D<1/2$ is unique and satisfies $\deg C^D=ha(G)$, the proof filtering $C$ by height-one quotients and using the Oort–Tate classification of finite flat group schemes of order $p$. Existence itself, however, is not re-proved: it is imported from Fargues's theorem with its restriction $p>3$, and the whole refined machinery rests on Hypothesis 0.1. Two structural facts shape the open problem. First, by Theorem 3.2 the condition $ha(G)<1/2$ is not merely sufficient but necessary for a canonical subgroup in the paper's sense, so 'going beyond $1/2$' must mean covering small $p$ and weakening the sufficient bounds toward this necessary one, not literally removing the bound. Second, the paper's Remark 3.7 observes that if Hypothesis 0.1 fails then $ha(G)=1$ and no canonical subgroup can exist, so the hypothesis is harmless for the bare existence statement; the genuine generalization target is the partial-invariant theory for general signatures, where the eigenspace dimensions of $\omega_G\otimes K$ are not constant — the regime of the $\mu$-ordinary locus (the locus where the Newton polygon of $G$, constrained by the $\mathcal{O}_F$-action, is as low as possible), which is the best substitute for the ordinary locus when the latter is empty. For ramified $F$, the standard substitute for the freeness demanded by Hypothesis 0.1 is the Pappas–Rapoport condition — introduced by Pappas and Rapoport for Hilbert modular varieties with ramified $F$ — which asks that $\omega_G$ carry a filtration with locally free graded pieces of prescribed ranks.

Problem Statement

Let $p$ be a prime, $F/\mathbb{Q}_p$ a finite extension with ramification index $e$ and residue degree $f$, $\mathcal{O}_F$ its ring of integers with uniformizer $\pi$ (a generator of its maximal ideal), and $K$ a complete valued extension of $\mathbb{Q}_p$ containing the Galois closure of $F$, with valuation normalized by $v(p)=1$. Let $G$ be a $p$-divisible group over $\mathcal{O}_K$ endowed with an action of $\mathcal{O}_F$ (with no assumption that $\omega_G\otimes_{\mathcal{O}_K}K$ is free over $K\otimes_{\mathbb{Z}_p}\mathcal{O}_F$, i.e. without Hypothesis 0.1, so general signatures of the action are admitted).

The problem is to construct the canonical subgroup $C\subset G[p]$ — the finite flat subgroup of height $\dim G$ whose Cartier dual has degree strictly less than $1/2$ (unique when it exists, and shown in the source to lift the kernel of the Frobenius modulo $p^{1-ha(G)}$) — together with its $\mathcal{O}_F$-stable pieces $C[\pi^k]$ and the canonical subgroups of the successive quotients $G/C[\pi^k]$, and to compute their degrees and partial degrees and the (partial) Hasse invariants of the quotients, under hypotheses as weak as possible on the Hasse invariant. Concretely:

(i) Existence without residue-characteristic restriction: prove (or disprove) existence of the canonical subgroup of $G$ and of $G/C[\pi^k]$ for $p=2$ and $p=3$ in the $\mathcal{O}_F$-action setting under Hasse-invariant hypotheses as weak as possible — the currently known sufficient bounds being those of Fargues's theorem ($ha<1/2$ for $p>3$, $ha<1/3$ for $p=3$, $ha<1/(2p)$ and $1/(3p)$ for the quotients) and of the source's partial-degree formulas ($ha<\min(1/e,1/2)$, $1/(pe)$) — noting that, in the source's normalization, existence already forces $ha(G)=\deg C^D<1/2$, so the target is not removal of the $1/2$ barrier itself.

(ii) Partial invariants without Hypothesis 0.1: extend the definitions of partial Hasse invariants and partial degrees to general signatures where the eigenspace dimensions of $\omega_G\otimes K$ are not constant — including the case of ramified $F$ — replacing Hypothesis 0.1 by whatever minimal structure makes the invariants well defined, and proving their independence of auxiliary choices such as orderings of the embeddings. Since failure of Hypothesis 0.1 forces $ha(G)=1$ and hence precludes the classical canonical subgroup at general signatures (the source's Remark 3.7), the target there is a refined invariant theory — for instance, partial degrees of a canonical filtration in the $\mu$-ordinary regime — which, when Hypothesis 0.1 holds and $ha(G)$ satisfies the source's smallness bounds, specializes to the source's formulas relating the partial degrees of $C[\pi^k]/C[\pi^{k-1}]$ and the partial Hasse invariants of $G/C[\pi^k]$ to the partial Hasse invariants of $G$.

The overarching objective is a canonical-subgroup theory for $p$-divisible groups with $\mathcal{O}_F$-action (existence, uniqueness, and degree bookkeeping for the subgroup, its $\pi$-power filtration, and the successive quotients) valid for all residue characteristics, together with a well-defined partial-invariant theory beyond Hypothesis 0.1 at general signatures.

References

1. Stéphane Bijakowski, Partial Hasse Invariants, Partial Degrees, and the Canonical Subgroup, Canadian Journal of Mathematics 70 (2018), 742–772, DOI 10.4153/cjm-2016-052-8, arXiv:1508.07604, https://arxiv.org/abs/1508.07604
2. Laurent Fargues, La filtration canonique des points de torsion des groupes p-divisibles, Annales scientifiques de l'École normale supérieure 44 (2011), 905–961, DOI 10.24033/asens.2157, https://doi.org/10.24033/asens.2157
3. Yichao Tian, Canonical subgroups of Barsotti-Tate groups, Annals of Mathematics 172 (2010), 955–988, DOI 10.4007/annals.2010.172.955, https://doi.org/10.4007/annals.2010.172.955
4. Shin Hattori, Canonical subgroups via Breuil-Kisin modules for p=2 (2011), arXiv:1109.4212, https://arxiv.org/abs/1109.4212
5. Joseph Rabinoff, Higher-level canonical subgroups for p-divisible groups (2009), arXiv:0910.3323, https://arxiv.org/abs/0910.3323
6. Vincent Pilloni, Benoit Stroh, Partial canonical subgroups (2009), arXiv:0907.1682, https://arxiv.org/abs/0907.1682
7. Peter Scholze, On torsion in the cohomology of locally symmetric varieties, Annals of Mathematics 182 (2015), 945–1066, DOI 10.4007/annals.2015.182.3.3, https://doi.org/10.4007/annals.2015.182.3.3
8. Valentin Hernandez, La filtration canonique des O-modules p-divisibles, Mathematische Annalen 375 (2019), 17–92, DOI 10.1007/s00208-018-1789-2, arXiv:1611.07396, https://arxiv.org/abs/1611.07396
9. Stéphane Bijakowski, Duality, refined partial Hasse invariants and the canonical filtration, Mathematical Research Letters 25 (2018), 1109–1142, DOI 10.4310/mrl.2018.v25.n4.a3, arXiv:1612.05078, https://arxiv.org/abs/1612.05078
10. Stéphane Bijakowski, Valentin Hernandez, Groupes p-divisibles avec condition de Pappas-Rapoport et invariants de Hasse, Journal de l'École polytechnique — Mathématiques 4 (2017), 935–972, DOI 10.5802/jep.60, arXiv:1611.10110, https://arxiv.org/abs/1611.10110
11. Davide A. Reduzzi, Liang Xiao, Partial Hasse invariants on splitting models of Hilbert modular varieties, Annales scientifiques de l'École normale supérieure 50 (2017), 579–607, DOI 10.24033/asens.2328, https://doi.org/10.24033/asens.2328
12. Stéphane Bijakowski, Andrea Marrama, The integral Hodge polygon for p-divisible groups with endomorphism structure, Tunisian Journal of Mathematics 6 (2024), 189–224, DOI 10.2140/tunis.2024.6.189, arXiv:2303.06166, https://arxiv.org/abs/2303.06166
13. Andrea Marrama, Hodge–Newton filtration for p-divisible groups with ramified endomorphism structure, Documenta Mathematica 27 (2022), 1805–1863, DOI 10.4171/dm/x19, arXiv:2010.13293, https://arxiv.org/abs/2010.13293
14. Rebecca Bellovin, Hanlin Cai, Sean Howe, Tongmu He, Characterizing perfectoid covers of abelian varieties (2025), arXiv:2501.03974, https://arxiv.org/abs/2501.03974