Full level structures on finite flat group schemes beyond rank-$p$ squares: the cases $G^n$ with $n\geq 3$, stack-level naturality, and higher height

Guan (2021) constructed full level structures -- closed subschemes of the homomorphism scheme $\mathrm{Hom}_S((\mathbb{Z}/p^r\mathbb{Z})^g, H)$ that are flat of rank $|\mathrm{GL}_g(\mathbb{Z}/p^r\mathbb{Z})|$, agree with the scheme of isomorphisms away from $p$, and are stable under base change -- for squares $G^2$ of a rank-$p$ Oort--Tate group scheme and for squares of height-one truncated $p$-divisible groups, and proved that no construction satisfying all desired axioms, including naturality under all group scheme isomorphisms, exists on the Oort--Tate product stack. The completion of this theory is open on three linked fronts: constructing full level structures on $G^n$ for $n\geq 3$ (the natural candidate condition overcounts, giving rank 169 over $\mathbb{F}_2$ versus $|\mathrm{GL}_3(\mathbb{F}_2)|=168$ and 11473 over $\mathbb{F}_3$ versus 11232), determining which weakened naturality axioms permit a stack-level notion, and extending the construction to truncated $p$-divisible groups of height greater than one. These three cores have not been addressed in the literature.

Background

A finite flat commutative group scheme $H$ over a base scheme $S$ is a group scheme whose structure sheaf $\mathcal{O}_H$ is a locally free $\mathcal{O}_S$-algebra of finite rank; its rank is that module rank. In characteristic $p$, the $p$-torsion of an abelian variety is such a group scheme but is generally not étale (étale means infinitesimally trivial, so that over an algebraically closed field the closed points form a constant finite group), so the classical notion of full level structure -- an isomorphism $(\mathbb{Z}/p^r\mathbb{Z})^g \xrightarrow{\sim} H$ over bases where $p$ is invertible -- has no direct integral analogue. Katz and Mazur, following Drinfeld, defined full level structures on group schemes embeddable in a curve via 'full sets of sections' (the sections generate $H$ as Cartier divisors, i.e. the sum of their associated effective Cartier divisors equals the pullback of the identity section), which suffices for modular curves; their suggested generalization via '$\times$-homomorphisms' (the same style of Cartier-divisor condition imposed on homomorphisms from a constant group scheme, defined without an embedding into a curve) fails flatness, as observed by Chai and Norman, already on $\mu_p \times \mu_p$ (here $\mu_p$ is the group scheme of $p$-th roots of unity, the kernel of $t \mapsto t^p$ on the multiplicative group $\mathbb{G}_m$). Wake then treated $H = \mu_p \times \mu_p$ over $\mathrm{Spec}\, \mathbb{Z}$ by requiring that all nonzero linear combinations of the rows and columns of the universal homomorphism be 'primitive', where a primitive element of a finite flat group scheme (in the sense of Kottwitz--Wake) is a point lying in the closed subscheme cut out by the annihilator of the augmentation ideal (the kernel of the counit $\mathcal{O}_H \to \mathcal{O}_S$); for the Oort--Tate scheme $\mathrm{Spec}\, A[x]/(x^p - tx)$ this is $x^{p-1} = t$. The Oort--Tate theorem classifies all finite flat commutative group schemes of rank $p$ over a $\mathbb{Z}_p$-scheme: locally they are $\mathrm{Spec}\, A[x]/(x^p - ux)$ with $uv = w_p$ a fixed unit multiple of $p$, and the moduli stack $OT$ of such group schemes is the quotient stack $[(\mathrm{Spec}\, \mathbb{Z}_p[s,t]/(st - w_p))/\mathbb{G}_m]$ (the stack encoding the $\mathbb{G}_m$-orbits of this parametrization). A $p$-divisible group (Barsotti--Tate group) is an inductive system of finite flat group schemes of ranks $p^{nh}$ for a fixed height $h$; its truncations $G[p^r]$ are the finite stages, and 'height one, rank $p$' is exactly the Oort--Tate case. Cartier duality sends a finite flat commutative group scheme $H$ to its character group scheme $\mathrm{Hom}(H, \mathbb{G}_m)$, exchanging for example $\mu_p$ with the constant group scheme $(\mathbb{Z}/p\mathbb{Z})$. Guan generalized Wake's rows-and-columns construction to $G^2$ for any Oort--Tate $G$ over a $\mathbb{Z}_p$-scheme $S$ and to squares of height-one truncated $p$-divisible groups, obtaining closed subschemes $\mathrm{Hom}^*_S((\mathbb{Z}/p^r\mathbb{Z})^2, G^2)$ -- where $\mathrm{Hom}_S((\mathbb{Z}/p^r\mathbb{Z})^2, G^2)$ is the scheme of homomorphisms of group schemes over $S$ from the constant group scheme $(\mathbb{Z}/p^r\mathbb{Z})^2$, canonically isomorphic to $G^4$ as an $S$-scheme -- flat over $S$ of the expected rank $|\mathrm{GL}_2(\mathbb{Z}/p^r\mathbb{Z})|$. The expected axioms for such a full level structure on a group scheme $H$ of rank $p^l$ with generic fiber étale-locally isomorphic to $(\mathbb{Z}/p^r\mathbb{Z})^g$ are: (1) $\mathrm{Hom}^*_S((\mathbb{Z}/p^r\mathbb{Z})^g, H)$ is flat over $S$ of rank $|\mathrm{GL}_g(\mathbb{Z}/p^r\mathbb{Z})|$; (2) it is invariant under the right $\mathrm{GL}_g(\mathbb{Z}/p^r\mathbb{Z})$-action and, after inverting $p$, coincides with the scheme of isomorphisms $(\mathbb{Z}/p^r\mathbb{Z})^g \to H$; (3) it is compatible with arbitrary base change; (4) it is natural under every isomorphism of group schemes. Guan's Theorem 6.1 proves that (4) is impossible to satisfy in general: on the product stack $OT \times OT$ no fibered functor of full level structures satisfying (1)--(4) exists, the obstruction being that the unique possible fiberwise construction is not invariant under the action of $\mathrm{GL}_2(\overline{\mathbb{F}}_p) = \mathrm{Aut}_{\overline{\mathbb{F}}_p}(\alpha_p \times \alpha_p)$, Wake's example being taken over $\overline{\mathbb{F}}_p$ with $p > 2$ (here $\alpha_p$ is the group scheme given as the kernel of the Frobenius map $t \mapsto t^p$ on the additive group $\mathbb{G}_a$, and $\mathrm{Aut}$ denotes the automorphism group scheme); the counterexample is due to Wake, who with Kottwitz had found the nonextendability in unpublished work. The motivation throughout is the construction of integral models of Shimura varieties (Shimura varieties generalize modular curves, the moduli spaces of elliptic curves, to higher dimensions) with full level structure at primes dividing the level, where a flat level structure on the $p$-torsion group scheme is exactly what is needed.

Problem Statement

Let $p$ be a prime and $S$ a $\mathbb{Z}_p$-scheme. For a finite flat commutative group scheme $H$ over $S$ of rank $p^l$ that is annihilated by $p^r$ and whose generic fiber is étale-locally isomorphic to $(\mathbb{Z}/p^r\mathbb{Z})^g$, a full level structure should be a closed subscheme $\mathrm{Hom}^*_S((\mathbb{Z}/p^r\mathbb{Z})^g, H)$ of $\mathrm{Hom}_S((\mathbb{Z}/p^r\mathbb{Z})^g, H) \cong H^g$ satisfying: (1) flatness over $S$ of rank $|\mathrm{GL}_g(\mathbb{Z}/p^r\mathbb{Z})|$; (2) invariance under the right $\mathrm{GL}_g(\mathbb{Z}/p^r\mathbb{Z})$-action, with agreement over $S[1/p]$ with the scheme of isomorphisms $(\mathbb{Z}/p^r\mathbb{Z})^g \to H$; (3) compatibility with arbitrary base change $T \to S$; and (4) naturality under isomorphisms of group schemes. Guan constructed such structures satisfying (1)--(3) -- natural under isomorphisms $G^2 \to G'^2$ of the form $\varphi \oplus \varphi$, but not under all isomorphisms -- for $G^2$ with $G$ an Oort--Tate group scheme of rank $p$ and for squares of height-one truncated $p$-divisible groups, and proved (Theorem 6.1) that no construction satisfying all of (1)--(4) exists over the product stack $OT \times OT$. The problem, sharing the single objective of completing the theory of full level structures on finite flat commutative group schemes in characteristic $p$ (as needed for integral models of Shimura varieties with full level at $p$), has three linked open cores: (a) for an Oort--Tate group scheme $G$ over $S$, construct a full level structure $\mathrm{Hom}^*_S((\mathbb{Z}/p\mathbb{Z})^n, G^n)$ satisfying (1)--(3) for $n \geq 3$, at least for $n = 3$: the natural candidate, requiring that after left and right $\mathrm{GL}_n(\mathbb{F}_p)$-actions and possibly Cartier dual every $2 \times 3$ block give a partial level structure, yields closed subschemes of ranks 169 over $\mathbb{F}_2$ and 11473 over $\mathbb{F}_3$, exceeding the required $|\mathrm{GL}_3(\mathbb{F}_2)| = 168$ and $|\mathrm{GL}_3(\mathbb{F}_3)| = 11232$, so the further conditions cutting out exactly the right rank, while preserving flatness and base-change compatibility, must be found; (b) determine which weakenings of the naturality axiom (4) -- for example, naturality only under isomorphisms $G^2 \to G'^2$ of the form $\varphi \oplus \varphi$, under which Guan's fiberwise construction does behave well -- admit a canonical stack-level full level structure on $OT \times OT$ or on stacks of finite flat commutative group schemes of a specified type, or, for a specified weakening, establish that it still conflicts with the $\mathrm{GL}_2(\overline{\mathbb{F}}_p)$-noninvariance counterexample of Theorem 6.1; (c) extend the construction beyond height one, that is, define full level structures on truncations $G[p^r]$ of $p$-divisible groups of height greater than one (or on more general finite flat commutative group schemes) satisfying (1)--(3), where Guan notes that new ideas are required. The three cores are separately posed open problems: none is subsumed by another, so a resolution of one does not by itself resolve the others, the objective stated above being the completion of the theory across all three fronts; correspondingly, in (b) the aim is the classification of the weakenings of the naturality axiom (4), and in (c) the aim is every height $h > 1$.

References

1. Chuangtian Guan, Full Level Structure On Some Group Schemes, Research in Number Theory 7 (2021), no. 2, Art. 34, DOI 10.1007/s40993-021-00260-2, https://doi.org/10.1007/s40993-021-00260-2 (arXiv:2008.12400, https://arxiv.org/abs/2008.12400)
2. Preston Wake, Full level structures revisited: pairs of roots of unity, Journal of Number Theory 168 (2016), 81-100, DOI 10.1016/j.jnt.2016.04.011, https://doi.org/10.1016/j.jnt.2016.04.011 (arXiv:1506.01750, https://arxiv.org/abs/1506.01750)
3. Robert Kottwitz and Preston Wake, Primitive elements for p-divisible groups, Research in Number Theory 3 (2017), Art. 20, DOI 10.1007/s40993-017-0084-8, https://doi.org/10.1007/s40993-017-0084-8 (arXiv:1510.02814, https://arxiv.org/abs/1510.02814)
4. John Tate and Frans Oort, Group schemes of prime order, Annales Scientifiques de l'École Normale Supérieure (4) 3 (1970), 1-21, DOI 10.24033/asens.1186, https://doi.org/10.24033/asens.1186
5. Nicholas M. Katz and Barry Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985, DOI 10.1515/9781400881710, https://doi.org/10.1515/9781400881710