A study of perfectoid rings via Galois cohomology

Abstract For a connected reductive group $G$ over a local or global field $K$, we define a diamond (or power) operation $$ \begin{align*} &(\xi,n)\mapsto \xi^{\Diamond n}\,\colon\, \mathrm{H}^1\kern -0.8pt(K,G)\times{\mathbb Z}\to \mathrm{H}^1\kern -0.8pt(K,G)\end{align*} $$ of raising to power $n$ in the Galois cohomology pointed set. This operation is new when $K$ is a number field. We show that this power operation has many good properties. When $G$ is a torus, the set $\mathrm{H}^{1}\kern -0.8pt(K,G)$ has a natural group structure, and $\xi ^{\Diamond n}$ then coincides with the $n$-th power of $\xi $ in this group. On the other hand, we show that a power operation on $\mathrm{H}^{1}\kern -0.8pt(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using the power operation, for a cohomology class $\xi $ in $\mathrm{H}^{1}\kern -0.8pt(K,G)$ over local or global field, we define the period $\operatorname{per}(\xi )$ to be the least integer $n\geqslant 1$ such that $\xi ^{\Diamond n}=1$. We define the index $\operatorname{ind}(\xi )$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $\xi $. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ over a local or global field $K$, we show that $\operatorname{per}(\xi )$ divides $\operatorname{ind}(\xi )$ and that $\operatorname{ind}(\xi )$ may be strictly greater than $\operatorname{per}(\xi )$, but they always have the same prime factors.

Transfer principles for Galois cohomology and Serre's conjecture II

We generalise a result of Fourquaux and Xie thereby completely determining the relationship between ℚ p -analytic and L-analytic Lie algebra cohomology of analytic (φ L ,Γ L )-modules. We use the results to conclude that for L≠ℚ p , there exist examples of étale (φ L ,Γ L )-modules over Robba rings whose ℚ p -analytic cohomology does not arise as a base change of Galois cohomology.

This study investigates the asymptotic behavior of the ranks of Selmer groups associated with elliptic curves possessing a rational 2-torsion point defined over the integers. The Selmer group plays a central role in understanding the Mordell–Weil group and the Birch and Swinnerton-Dyer conjecture. The arithmetic of elliptic curves with torsion points has long attracted significant interest, with foundational results tracing back to the work of Mordell, Selmer, and later refinements by Cassels and others. In particular, the behavior of 2-Selmer groups provides insights into the distribution of ranks and the structure of rational points. Building upon previous methods developed for quadratic twists and leveraging tools from Galois cohomology, we demonstrate that the upper bounds on the size of these Selmer groups are unbounded within certain infinite families of elliptic curves. Our approach highlights the interplay between local conditions at primes and global properties of the curve, offering new perspectives on how torsion influences Selmer ranks.

Let L/\mathbb{Q}_{p} be a finite extension. We introduce L -typical prisms , a mild generalization of prisms. Following ideas of Bhatt, Scholze, and Wu, we show that certain vector bundles, called Laurent F -crystals, on the L -typical prismatic site of a formal scheme X over \operatorname{Spf}\mathcal{O}_{L} are equivalent to \mathcal{O}_{L} -linear local systems on the generic fiber X_{\eta} . We also give comparison theorems for computing the étale cohomology of a local system in terms of the cohomology of its corresponding Laurent F -crystal. In the case X=\operatorname{Spf}\mathcal{O}_{K} for K/L a p -adic field, we show that this recovers the Kisin–Ren equivalence between Lubin–Tate (\varphi_{q},\Gamma) -modules and \mathcal{O}_{L} -linear representations of G_{K} , as well as the results of Kupferer and Venjakob for computing Galois cohomology in terms of Herr complexes of (\varphi_{q},\Gamma) -modules. We can thus regard Laurent F -crystals on the L -typical prismatic site as providing a suitable notion of relative (\varphi_{q},\Gamma) -modules.

Abstract It was shown by the first author that, over an algebraically closed field of characteristic zero, every fusion 2-category is Morita equivalent to a connected fusion 2-category, that is, one arising from a braided fusion 1-category. This result has recently allowed for a complete classification of fusion 2-categories. Here we establish that compact semisimple tensor 2-categories, which generalize fusion 2-categories to an arbitrary field of characteristic zero, also enjoy this “Morita connectedness” property. In order to do so, we generalize to an arbitrary field of characteristic zero many well-known results about braided fusion 1-categories over an algebraically closed field. Most notably, we prove that the Picard group of any braided fusion 1-category is indfinite, generalizing the classical fact that the Brauer group of a field is torsion. As an application of our main result, we derive the existence of braided fusion 1-categories indexed by the fourth Galois cohomology group of the absolute Galois group that represent interesting classes in the appropriate Witt groups.

Let [Formula: see text] be a smooth projective rational variety carrying a regular action of a finite abelian group [Formula: see text]. We give examples of effective computation of the Brauer group of the quotient stack [Formula: see text] in dimensions [Formula: see text] and [Formula: see text] using residues in Galois cohomology and the geometry of fixed loci. In particular, we compute [Formula: see text] for all [Formula: see text]-minimal del Pezzo surfaces.

Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is “Yes” when K has no real embeddings. We show that otherwise the answer is “No”. Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets H1(K,G) for all reductive K-groups G in a functorial way.

Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of G S (k), the Galois group of the maximal extension of a global field k that is unramified outside a finite set S of places, as k varies among a certain family of extensions of a fixed global field Q.We define a group B S (k, A), for each finite simple G S (k)-module A, to generalize the work of Koch and Shafarevich on the pro- completion of G S (k).We prove that G S (k) always admits a balanced presentation when it is finitely generated.In the setting of the nonabelian Cohen-Lenstra heuristics, we prove that the unramified Galois groups studied by the Liu-Wood-Zureick-Brown conjecture always admit a balanced presentation in the form of the random group in the conjecture.

Abstract Let $p$ be a prime number and let $F$ be a field of characteristic different from $p$ . We prove that there exist a field extension $L/F$ and $a,b,c,d$ in $L^{\times }$ such that $(a,b)=(b,c)=(c,d)=0$ in $\mathrm {Br}(L)[p]$ but the mod p Massey product $\langle a,b,c,d\rangle$ is not defined over $L$ . Thus, the strong Massey vanishing conjecture at the prime $p$ fails for $L$ , and the cochain differential graded ring $C^{* }(\Gamma _L,\mathbb Z/p\mathbb Z)$ of the absolute Galois group $\Gamma _L$ of $L$ is not formal. This answers a question of Positselski. As our main tool, we define a secondary obstruction that detects non-triviality of unramified torsors under tori, and which is of independent interest.

Abstract In [2], Pillay introduced definable Galois cohomology, a model-theoretic generalization of Galois cohomology. Let M be an atomic and strongly $\omega $ -homogeneous structure over a set of parameters A. Let B be a normal extension of A in M. We show that a short exact sequence of automorphism groups $1 \to \operatorname {\mathrm {Aut}}(M/B) \to \operatorname {\mathrm {Aut}}(M/A) \to \operatorname {\mathrm {Aut}}(B/A) \to 1$ induces a short exact sequence in definable Galois cohomology. We also discuss compatibilities with [3]. Our result complements the long exact sequence in definable Galois cohomology developed in [4].

Galois cohomology of elliptic curves over anticyclotomic extensions

The Brauer group, an important concept in algebra, plays a pivotal role in the study of central simple algebras and their classification. This paper explores the algebraic structures underlying the Brauer group, emphasizing its connections with division algebras, Galois cohomology, and class field theory. We delve into the properties and operations that define the Brauer group, examining how these properties extend across various algebraic and geometric contexts. Additionally, Brauer diagrams, visual representations of elements in the Brauer group, are studied for their applications in tensor categories, quantum algebra, and knot theory. Through a combination of theoretical analysis and illustrative examples, this work aims to provide a comprehensive understanding of the interplay between the Brauer group and Brauer diagrams, shedding light on their significance in modern mathematical research.

Herr complex of (φ,τ)-modules

Abstract Let G be the indefinite unitary group U(p, p), H ≃ GL p (ℂ) its symmetric subgroup, P S the Siegel parabolic subgroup of G, and B H a Borel subgroup of H. In this article, we give a classification of the orbit decomposition H \(H/B H × G/P S ) of the real double flag variety by using the Galois cohomology in the case where p = 2.

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}$$\\end{document} with unitriangular Galois groups and contain the Legendre and Rédei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the qth roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient and relate it to Massey product elements in Galois cohomology.

Semisimple elements and the little Weyl group of real semisimple [formula omitted]-graded Lie algebras

We explain a construction of explicit extensions -of rational Hodge structures and of p-adic Galois representations -in a simple context: the cohomology of 1-{some points} relative to {some other points}.These extensions are naturally related to Dirichlet characters, and we connect the nonsplitting of these extensions to the values at s = 0 and s = 1 of associated Dirichlet L-functions L(s, ).We highlight the close parallels between the proofs of nonsplitting in both the Hodge-theoretic and p-adic cases, emphasizing the use of de Rham theory.We also indicate connections with Euler systems along with variations on these constructions in the setting of modular curves.This paper is intended as an introduction to some of the key ideas in forthcoming constructions of Galois cohomology classes and Euler systems in a range of settings.

We prove that a strengthened version of Minač–Tân’s Massey Vanishing Conjecture holds true for fields with a finite number of square classes whose maximal pro-2 Galois group is of elementary type (as defined by I. Efrat). In particular, this proves Minač–Tân’s Massey Vanishing Conjecture for Pythagorean fields with a finite number of square classes and their finite extensions.