Torsion structure of Hodge-Witt cohomology of abelian varieties in positive characteristic: the isogeny class of the unipotent part of the p-primary Brauer group

In 1958 Serre introduced the Witt-vector cohomology groups $H^q(X,\mathscr{W})$ of a variety $X$ over a perfect field of characteristic $p>0$. He proved that for an abelian variety $A$ the group $H^1(A,W\mathcal{O}_A)$ is free of finite rank over the Witt ring $W$, but his own counterexample, the product of two supersingular elliptic curves, showed that $H^2$ can fail to be finitely generated over $W$. Later work converted this failure into a precise structure theory: Illusie and Raynaud proved that the failure of finite generation of the Hodge-Witt groups $H^j(X,W\Omega^i)$ is exactly measured by torsion numbers $T^{ij}$, that the $V$-torsion submodule is, up to finite length, the Cartier module of a smooth connected unipotent formal group, and that ordinary varieties have all Hodge-Witt groups finitely generated. In modern terms, for an abelian variety $A$ of dimension $g$ the $p$-primary torsion of the Brauer group decomposes as $\mathrm{Br}(A)[p^\infty]\simeq(\mathbb{Q}_p/\mathbb{Z}_p)^{r-\rho}\oplus U(k)$, where $U$ is a connected commutative unipotent group with $\dim U=T^{0,2}=h^{0,2}-m^{0,2}$ (Crew, extended to higher dimensions by Ekedahl). The fine structure of this torsion is the surviving open core: the isogeny class of $U$, that is, the multiset of Witt groups $W_{n_i}$ in its Serre isogeny decomposition, was determined for abelian varieties of dimension at most $3$ by Yang (2025, for $p\neq 2$) but is open in general; the most recent treatment (Grammatica-Skorobogatov-Yang, 2025) proves that the $p$-exponent of $U$ is at most $g-1$ for $p\neq 2$, sharply, and poses this classification as its main open question. For supergeneral abelian varieties of dimension $g$, Yang conjectures that $U$ is isogenous to $\prod_{i=1}^{g-1}W_i$; the conjectured value is realized by an exhibited supergeneral abelian variety in each dimension, but no proof or counterexample is known for the general statement, and the precise structure of $U$ up to isomorphism, even for supergeneral abelian threefolds, likewise remains open.

Background

Let $k$ be a perfect field of characteristic $p>0$, and let $W=W(k)$ be its ring of $p$-adic Witt vectors, the complete discrete valuation ring with residue field $k$ and uniformizer $p$. For a smooth projective variety $X$ over $k$, Serre (1958) introduced the Witt-vector cohomology groups $H^q(X,\mathscr{W})$, the sheaf cohomology of the sheaf of Witt vectors, obtained as the inverse limit over $n$ of the cohomology of the sheaves $W_n\mathcal{O}_X$ of length-$n$ truncated Witt vectors. These groups are a characteristic-$p$ analogue of $p$-adic cohomology. Illusie (1979) embedded them into the de Rham-Witt complex $W\Omega_X^\bullet$, a system of differential graded algebras lifting the de Rham complex to all Witt-vector lengths; the cohomology groups $H^j(X,W\Omega^i)$ of its individual graded pieces are called Hodge-Witt cohomology, and $H^q(X,\mathscr{W})$ is the case $i=0$. The groups $H^j(X,W\Omega^i)$ carry semilinear Frobenius and Verschiebung operators $F$ and $V$, and they form the $E_1$ page of the slope spectral sequence converging to crystalline cohomology $H^\bullet_{\mathrm{cris}}(X/W)$, the $p$-adic cohomology theory constructed by Berthelot and Grothendieck which for varieties that do not lift to characteristic zero plays the role that singular cohomology plays over the complex numbers.

Serre proved that for an abelian variety $A$ over $k$ the group $H^1(A,W\mathcal{O}_A)$ is a free $W$-module of finite rank (his result is reproduced as Proposition 2.19 in Illusie's 1979 paper), and that the ordinary Hodge cohomology $H^\bullet(A,\mathcal{O}_A)$ is an exterior algebra on $H^1(A,\mathcal{O}_A)$. In the same paper he showed, however, that finite generation fails in degree $2$: if $E_1$ and $E_2$ are elliptic curves over $k$ with Hasse invariant $0$, equivalently supersingular elliptic curves (elliptic curves whose formal Brauer group has the maximal height $2$, or equivalently whose Newton polygon is a single segment of slope $1/2$), then for the abelian surface $X=E_1\times E_2$ the group $H^2(X,W\mathcal{O}_X)$ is not a finitely generated $W$-module. Illusie and Raynaud (1983) later established the general facts behind this phenomenon: $H^j(X,W\Omega^i)$ is finitely generated over $W$ modulo its $V$-torsion, the $V$-torsion submodule is annihilated by a power of $p$, and it is, up to a module of finite length, the Cartier module of a smooth connected unipotent formal group (the Cartier module of a formal group is the $W$-module equipped with the Verschiebung operator $V$ that is equivalent to the formal group); moreover $H^j(X,W\Omega^i)$ is finitely generated over $W$ if and only if the torsion number $T^{ij}$, the invariant measuring its $V$-torsion, vanishes, and a variety is called Hodge-Witt when all these groups are finitely generated. They also proved that ordinary varieties (those whose Hodge-de Rham spectral sequence degenerates with Frobenius acting bijectively on each $H^j(X,\Omega^i)$; for abelian varieties this is equivalent to the p-rank being maximal) are Hodge-Witt, which affirmatively settles the finiteness question under ordinary hypotheses. Crew (1985), in work extended to higher dimensions by Ekedahl, made the size of the torsion numerical: writing $m_{ij}$ for the Hodge-Newton numbers, the slope-weighted multiplicities $m_{ij}=\sum_{i\leq\lambda<i+1}(i+1-\lambda)m_\lambda+\sum_{i-1\leq\lambda<i}(\lambda-i+1)m_\lambda$ of the crystalline Frobenius slopes, where $m_\lambda$ is the multiplicity of the slope $\lambda$ of $H^{i+j}_{\mathrm{cris}}$ (in the degree-two case this is $m^{0,2}=\sum_{0\leq\lambda<1}(1-\lambda)m_\lambda$), the dimension of the unipotent part is the difference $h^{ij}-m_{ij}$ of the Hodge number and the Hodge-Newton number, and for surfaces $T^{0,2}$ is upper semicontinuous in smooth proper families.

The modern form of the question passes to the Brauer group. The Brauer group $\mathrm{Br}(X)$ is the torsion part of the flat (fppf) cohomology group $H^2_{\mathrm{fl}}(X,\mathbb{G}_m)$, and its $p$-primary part in characteristic $p$ is governed by Hodge-Witt cohomology through the formal Brauer group, the Artin-Mazur formal group (the formal group pro-representing the functor sending an Artinian local $k$-algebra $R$ to $\ker(H^2(X\otimes R,\mathbb{G}_m)\to H^2(X,\mathbb{G}_m))$) and its flat variant, whose Cartier module is $H^2(X,W\mathcal{O}_X)$. For an abelian variety $A$ of dimension $g$ over an algebraically closed field $k$, one has a decomposition $\mathrm{Br}(A)[p^\infty]\simeq(\mathbb{Q}_p/\mathbb{Z}_p)^{r-\rho}\oplus U(k)$, where $r$ is the multiplicity of slope $1$ in $H^2_{\mathrm{cris}}(A/W)$, $\rho$ is the Picard number (the rank of the Néron-Severi group), and $U$ is a connected commutative unipotent algebraic group over $k$ with $\dim U=T^{0,2}=h^{0,2}-m^{0,2}$ and $h^{0,2}=\binom{g}{2}$ (Yang, Theorem 2.4.2, building on the work of Crew and Ekedahl). A unipotent group is an algebraic group admitting a filtration with successive quotients isomorphic to the additive group $\mathbb{G}_a$; a theorem of Serre says that every connected commutative unipotent group $U$ in characteristic $p$ is isogenous to (surjects onto, with finite kernel) a product of Witt vector groups $\prod_i W_{n_i}$, where $W_n$ is the group of length-$n$ Witt vectors, and the multiset of integers $n_i$, called the isogeny class of $U$, is uniquely determined by, and determines, the dimensions of the finite subgroup schemes $U[p^n]$ killed by $p^n$ for all $n$. Determining the isogeny class of $U$ therefore amounts to computing the integers $\dim U[p^n]$.

Concrete tools are available. A formula of Skorobogatov (proved in Yang's thesis for $p\neq 2$) identifies $\mathrm{Br}(A)[p^n]$ with the symmetric part $\mathrm{Hom}(A[p^n],A^\vee[p^n])^{\mathrm{sym}}$ of the homomorphism group between the $p^n$-torsion group schemes of $A$ and its dual $A^\vee$, modulo the image of $\mathrm{Hom}(A,A^\vee)^{\mathrm{sym}}$; and Yang's Theorem 4.1.2 computes $\dim U[p]$ from the Ekedahl-Oort type of $A$ (the elementary sequence classifying the Dieudonné module of $A[p]$, equivalently the isomorphism type of the $p$-torsion group scheme with its Verschiebung). Using these, and assuming $p\neq 2$, Yang classified the isogeny class of $U$ for all abelian threefolds: $U$ is trivial or a product of copies of $\mathbb{G}_a$ in all cases except the supergeneral ones, supersingular abelian threefolds with $a$-number $1$ (the $a$-number is $\dim_k\mathrm{Hom}(\alpha_p,A)$, where $\alpha_p$ is the kernel of Frobenius on $\mathbb{G}_a$), where $U$ is isogenous to $\mathbb{G}_a\times W_2$. The most recent treatment, by Grammatica, Skorobogatov, and Yang (2025), republishes the threefold classification, proves that the $p$-exponent of $U$ (the least $m$ with $p^mU=0$) is at most $g-1$ for $p\neq 2$, with equality for one exhibited supergeneral abelian variety in each dimension $g$, and shows that the isogeny class of $U$ is not an invariant of the isogeny class of $A$; the classification in dimension $g\geq 4$ and the structure of $U$ up to isomorphism remain open.

Problem Statement

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ of characteristic $p>0$, let $W=W(k)$, and let $U$ be the connected commutative unipotent algebraic group appearing in the decomposition $\mathrm{Br}(A)[p^\infty]\simeq(\mathbb{Q}_p/\mathbb{Z}_p)^{r-\rho}\oplus U(k)$ of the $p$-primary torsion of the Brauer group; equivalently, $U$ is the group whose Cartier module is, up to finite length, the $V$-torsion submodule of the Hodge-Witt cohomology group $H^2(A,W\mathcal{O}_A)$. The dimension of $U$ is known in general ($\dim U=T^{0,2}=h^{0,2}-m^{0,2}$, $h^{0,2}=\binom{g}{2}$), its $p$-exponent is known to be at most $g-1$ for $p\neq 2$ (a sharp bound), and the isogeny class of $U$ is known for $g\leq 3$ (for $p\neq 2$). The problem is to determine the fine structure of this torsion in general:

(1a) For the abelian varieties not already covered by the known $p\neq 2$ classification, namely those of dimension $g\geq 4$ (for any prime $p$) and those of dimension $g\leq 3$ in characteristic $p=2$, determine the isogeny class of $U$, i.e. the multiset of positive integers $(n_i)$ such that $U$ is isogenous to the product of Witt vector groups $\prod_i W_{n_i}$, as a function of standard invariants of $A$ (Newton polygon, Ekedahl-Oort type, $a$-number), or prove that the isogeny class is not determined by any function of these invariants; whether the isogeny class of $U$ is determined by such invariants at all is part of the question, since it is not an invariant of the isogeny class of $A$. Equivalently, compute the dimensions $\dim U[p^n]$ for all $n\geq 1$.

(1b) Prove or refute Yang's Conjecture 5.2.1: if $A$ is supergeneral (supersingular with $a$-number $1$) of dimension $g$, then $U$ is isogenous to $\prod_{i=1}^{g-1}W_i$; the conjectured value is known to hold for one exhibited supergeneral abelian variety in each dimension $g$, but no proof or counterexample is known in general. This sub-question is required independently of (1a): a proof that the standard invariants do not determine the isogeny class of $U$ does not settle the supergeneral case, and a proof of the supergeneral case does not settle the general classification.

(2) Determine the structure of $U$ up to isomorphism, not merely up to isogeny; this is open even for supergeneral abelian threefolds, where the isogeny class ($U$ is isogenous to $\mathbb{G}_a\times W_2$) is known but the precise structure is not.

More generally, in Hodge-Witt terms: for a non-ordinary (in particular supersingular) abelian variety $A$, determine the $V$-torsion submodule of each Hodge-Witt cohomology group $H^j(A,W\Omega^i)$ with $T^{ij}\neq 0$ as a module over the Dieudonné-type operators $F$ and $V$ on the Witt ring. The degree-$2$ case above, where the torsion is incarnated as the unipotent group $U$, is the central and most concretely formulated instance, and the task is to settle (1a) — the isogeny class of $U$ for the abelian varieties not already covered by the known $p\neq 2$ classification, namely dimension $g\geq 4$ for any prime $p$ and dimension $g\leq 3$ in characteristic $p=2$, whether in terms of Ekedahl-Oort-type data as Yang did for $g\leq 3$ (for $p\neq 2$) or by establishing that no such invariants suffice — together with (1b), Yang's Conjecture 5.2.1 in every dimension $g$, and (2), the structure of $U$ up to isomorphism; the determination of the $V$-torsion submodule of every $H^j(A,W\Omega^i)$ with $T^{ij}\neq 0$ is the general Hodge-Witt formulation of the same question, not a separate requirement.

References

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