p-Primary Torsion of the Brauer Group in Characteristic p

Technical Summary: $p$-Primary Torsion of the Brauer Group in Characteristic $p$

Problem Statement
The thesis investigates the structure of the $p$-primary component of the Brauer group, denoted $\text{Br}(X)[p^\infty]$, for a proper smooth variety $X$ over an algebraically closed field $k$ of characteristic $p > 0$. While the $\ell$-primary torsion for $\ell \neq p$ is well-understood via étale cohomology and the Kummer sequence, the $p$-primary part is significantly more subtle. The group $\text{Br}(X)[p^\infty]$ decomposes into a divisible part $(\mathbb{Q}_p/\mathbb{Z}_p)^{r-\rho}$ (where $r$ is the multiplicity of slope 1 in crystalline cohomology and $\rho$ is the Picard number) and a torsion part $H^3(X, \mathbb{Z}_p(1))[p^\infty]$. The central challenge addressed is the precise description of the latter, which fits into an exact sequence:
$$0 \to U(k) \to H^3(X, \mathbb{Z}_p(1))[p^\infty] \to J \to 0$$
where $U$ is a connected commutative unipotent algebraic group and $J$ is a finite group. The thesis aims to determine the dimension, isogeny class, and explicit structure of $U$ and $J$, particularly for abelian varieties.

Methodology
The research employs a synthesis of several advanced tools in arithmetic geometry:

1. Illusie's Theory of Logarithmic de Rham-Witt Sheaves: The thesis utilizes the de Rham-Witt complex $W\Omega^\bullet_{X/k}$ and its logarithmic subsheaves $W\Omega^i_{X/k, \log}$ to relate flat cohomology $H^i_{fppf}(X, \mu_{p^n})$ to crystalline cohomology.
2. Slope Spectral Sequences and Domino Numbers: The analysis relies on the slope spectral sequence $E_1^{pq} = H^q(X, W\Omega^p_{X/k}) \Rightarrow H^{p+q}_{\text{cris}}(X/W)$. The non-degeneration of this sequence at the $E_1$ page (due to $p$-torsion) is measured by "domino numbers" $T^{ij}$, introduced by Crew and Ekedahl. The dimension of the unipotent group $U$ is identified with the domino number $T^{0,2}$.
3. Dieudonné Module Theory: For abelian varieties, the thesis translates the geometric problem into linear algebra over the Cartier-Dieudonné ring $W\sigma[F, V]$. It employs the classification of finite commutative group schemes by Kraft and the stratification of moduli spaces by Ekedahl-Oort types.
4. Skorobogatov's Formula: A key methodological tool is a formula by Skorobogatov (for $p \neq 2$) that expresses $\text{Br}(A)[p^n]$ in terms of the symmetric homomorphisms $\text{Hom}(A[p^n], A^\vee[p^n])_{\text{sym}}$. This allows for explicit computation of the $p$-torsion subgroup $U[p]$ via the relations of Dieudonné modules.
5. Flat Duality and Formal Groups: The work applies Milne's flat duality theorem for surfaces and investigates the smoothness of flat Artin-Mazur formal groups without relying on Ekedahl's diagonal $t$-structure.

Key Contributions and Results

Part I: General Theory and First Applications

• Structure Theorem: The thesis confirms the split exact sequence $0 \to (\mathbb{Q}_p/\mathbb{Z}_p)^{r-\rho} \to \text{Br}(X)[p^\infty] \to H^3(X, \mathbb{Z}_p(1))[p^\infty] \to 0$.
• Unipotent Dimension: It establishes that the dimension of the unipotent group $U$ is given by the domino number $T^{0,2}$. For surfaces, this recovers Crew's formula: $\dim U = (h_{0,2} - h_{0,1}) - (m_{0,2} - m_{0,1})$.
• Ordinary Varieties: It is shown that if $X$ is ordinary, then $U = 0$ and $J$ is the Pontryagin dual of $\text{NS}(X)[p^\infty]$.
• Surfaces: For surfaces with $q = -p_a$ (including Enriques, Godeaux, and hyperelliptic surfaces), $U=0$. The finite group $J$ is identified as the dual of the $p$-torsion of the Neron-Severi group.
• Products: Using Ekedahl's Künneth formula, it is proven that if $X$ is ordinary, the unipotent part of $\text{Br}(X \times Y)$ is isomorphic to that of $\text{Br}(Y)$.

Part II: Abelian Varieties

• Skorobogatov's Formula Application: The thesis explicitly computes $\text{Br}(A)[p^n]$ for principally polarized abelian varieties using the symmetric homomorphism group.
• Dimension of $U[p]$: For a principally polarized abelian variety of dimension $g$ with Ekedahl-Oort type $\phi$, the dimension of the $p$-torsion subgroup $U[p]$ is calculated combinatorially based on the type.
• Classification for Threefolds: A complete classification of the isogeny class of $U$ for all abelian threefolds is provided. The results are tabulated by Newton polygon and $a$-number. A significant finding is that $U$ is not a product of additive groups $\mathbb{G}_a$ if and only if $A$ is supersingular with $a$-number 1 (termed "supergeneral"). In this specific case, $U$ is isogenous to $\mathbb{G}_a \times W_2$ (where $W_2$ is the Witt group of length 2).
• Explicit Hom Groups: For supergeneral principally polarized abelian threefolds, the connected component of $\text{Hom}(A[p^2], A^\vee[p^2])_{\text{sym}}$ is explicitly computed. It is shown to be a unipotent group $W(b,c)_2 \oplus \mathbb{G}_a$ that varies well-behavedly across the moduli space.
• Conjecture: A conjecture is proposed that for an arbitrary supergeneral abelian variety of dimension $g$, $U$ is isogenous to $\prod_{i=1}^{g-1} W_i$.

Part III: Further Topics and Applications

• Injectivity Criterion: The thesis generalizes a result of Ogus by proving an injectivity criterion for the map $d\log: H^2_{fppf}(X, \mu_{p^n}) \to H^2_{\text{cris}}(X/W_n)$ under the assumption that the Frölicher spectral sequence degenerates at $E_1$ and $H^1_{\text{cris}}$ is torsion-free. This applies to abelian varieties and K3 surfaces.
• Smoothness of Formal Groups: Without invoking Ekedahl's deep diagonal $t$-structure, the thesis establishes the smoothness of flat Artin-Mazur formal groups $\Phi^i_{fl}(A, \mathbb{G}_m)$ for all abelian threefolds and a large class of abelian fourfolds.
• Tate Conjecture Analogy: The final section discusses the Tate conjecture for divisors as an analogue of the Hodge conjecture, utilizing the Artin-Hasse exponential map and the structure of Witt vectors to frame the problem in terms of "uniformly finite" elements in crystalline cohomology.

Significance and Claims
The paper claims to provide a comprehensive structural description of the $p$-primary torsion of the Brauer group, moving beyond dimension formulas to determine the specific isogeny classes of the unipotent components.

• It resolves the structure of $\text{Br}(A)[p^\infty]$ for all abelian threefolds, identifying the unique case (supergeneral) where the unipotent group is non-trivial and not a product of $\mathbb{G}_a$.
• It offers explicit combinatorial formulas for the dimension of $U[p]$ based on Ekedahl-Oort types.
• It provides a new, elementary proof for the smoothness of certain formal groups, bypassing the need for the diagonal $t$-structure.
• The work proposes a conjectural description for the isogeny class of $U$ in the general supergeneral case, supported by explicit calculations in dimension 3 and partial results in higher dimensions.

The thesis emphasizes that while the dimension of $U$ was known via Crew and Ekedahl, the intricate behavior of unipotent groups in characteristic $p$ (specifically their isogeny classes) required the explicit Dieudonné module calculations and the application of Skorobogatov's formula presented herein.