Ramification filtration by moduli in higher-dimensional global class field theory

Let k be an algebraic number field (of finite degree) and K/k be a (finite) Galois extension with Galois group R. In connection with the class field theory, Weil [101 derived a certain canonical factor set a(o, r) of 9 in the idele-class group (K of K, which is uniquely determined by K/k in the sense of equivalence. For instance, a (mod R') -?f t r(o, A) (Ek k, essentially) (mod NK/k( RK)) is the Artin reciprocity law isomoprhism of 9/9' and Sk/NK/k( SK), where 9' is the commutator subgroup of R. A different approach to this (idele-class) factor set was given in [6]. Let, for each pair o, r, a(o, r) be a representative idele of a(o, r). Then (3a)(p, o, r) = a(o, r)a(po, r)-'a(p, o-r)a(p, oY)is, for each triple p, a, r, a principal idele, i.e. a non-zero element of K. We obtain thus a 3-cochain a(p, a, r) = (6a)(p, a, r) of 9 in the principal idele group PK of K, which is in fact a 3-cocycle. We saw, in [6], that this 3-cycocle a, or more precisely its cohomology class, attached in invariant fashion to the Galois extension K/k, has the order m/m', where m = (K: k) and m' is the least common multiple of the p-degrees of K/k, p running over all primes in k. It was shown further, in a joint paper [4] by Hochschild and the writer, that the cyclic group, of order m/m', generated by this 3-cohomology class is exactly the group of all 3-cohomology classes of 9 in PK with ceiling. This last group is in turn, according to EilenbergMacLane [1], nothing but the group of all Teichmtiller 3-cohomology classes for K/k, belonging to central simple algebras over K to which every element of ft can be extended (as automorphism). Now arises the problem to determine the exact algebra-class (though not unique) which possesses our a, attached to K/k, as its Teichmiiller-class. The present work is devoted to this problem and our solution is given in our Theorem in ?3. If a central simple algebra A over K, to which every automorphism of K/k can be extended, is given in a crossed-product form for instance, we can obtain its Teichmiiller-cocycle in a more or less explicit process involving the HilbertSpeiser theorem. On the other hand, in order to obtain our 3-cocycle a, we repeat the construction of the canonical idele-class factor set a given in [6], which is nothing but the global analogy of the local construction in Hochschild [3]1 and makes use of the idele interpretation of the principal genus theorem of Noether, but with more detailed analysis. Namely, we carry out all the processes in terms of ideles, rather than of idele-classes, preserving thus all principal idele factors which were discarded in [6]. This construction, involving the theorem of everywhere splitting algebras, the Hilbert-Speiser theorem and its idble analogy, leads to a. Though fairly explicit, these constructions of a and the Teichmfiller-

We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.

BackgroundQuaternions have emerged as powerful tools in higher-dimensional quantum mechanics as they provide homogeneous four-dimensional structure in quantum field theories, offer compact representations, and incorporate spin naturally. Quantum field theories then lead to the unification of fundamental interactions so the use of quaternion becomes necessary when we are dealing with higher-dimensional theories. On the other hand, supersymmetry is the theory of bosons and fermions and is an essential constituent of grand unified theories. The use of quaternion in supersymmetric field theories provides an excellent framework for higher-dimensional unification theories.ResultA complete theory for supersymmetric quaternionic quantum mechanics has been constructed for N = 1, 2, 4 supersymmetry in terms of one, two, and four supercharges and Hamiltonians, respectively. It has been shown that N = 4 SUSY is the quaternionic extension of the N = 2 complex SUSY and N = 1 real SUSY; also spin is the natural outcome of using quaternion units. Pauli and Dirac Hamiltonian and their relationship have also been obtained in quaternion space. It has been shown that quaternionic quantum mechanics are superior to ordinary and complex quantum mechanics because in the quaternion framework we do not need three different theories for N = 1,2,4 SQM but a single theory only.ConclusionsIt has been concluded that N = 1 real SUSY is equal to N = 2 complex SUSY which in turn is equal to N = 4 quaternion SUSY so one can arrive at higher-dimensional quantum field theories starting from lower-dimensional quantum theories. Higher-dimensional quaternion field theories are suitable for nonphotonic light cone particles which are not allowed in complex QFT, also noncommutative nature of quaternion gives an extra degree of freedom and may provide the possibility of some new particle, dark matter, or new phenomenon. Though quaternions provide an excellent framework in higher-dimensional field theories, there are certain challenges due to their noncommutativity as calculations become tedious where large terms are involved. Keeping in view the noble features of quaternion, we expect some development to get a better understanding of N = 8 supergravity, maximal supergravity (D = 11 − n), and maximal supersymmetry theories (N = 10) in terms of quaternion operators.

Let \(K\) be a global field. The aim of this speculative paper is to discuss the possibility of constructing the non-abelian version of global class field theory of \(K\) by “glueing” the non-abelian local class field theories of \(K_\nu \) in the sense of Koch, for each \(\nu \in \mathbb{h }_K\), following Chevalley’s philosophy of ideles, and further discuss the relationship of this theory with the global reciprocity principle of Langlands.

This paper advances the theory of Configurational Field Analysis (CFA) as a reconfiguration of Pierre Bourdieu’s field theory, aiming to address the limitations of Global Field Theory in analyzing the complexities of global and transnational phenomena. While the concept of the Global Field extended Bourdieu’s ideas to transnational and global arenas, it has been critiqued for its structural determinism, Eurocentrism, and its inability to fully capture the fluid, indeterminate, and contingent nature of global social dynamics. In response, this paper introduces social configurations as dynamic, relational constructs that emerge from specific historical and contextual conditions, rather than as fixed and universal structures. By integrating the concept of social configurations into field theory, CFA reconceptualizes social spaces as fluid and contested arenas where power, capital, and influence are continually negotiated. The paper proceeds by revisiting the foundations and critiques of Global Field Theory, followed by the introduction of social configurations and their theoretical advantages. Finally, it presents Configurational Field Analysis as a comprehensive framework, detailing its analytical steps and demonstrating its applicability to contemporary global issues. This framework not only addresses the methodological and analytical gaps in Global Field Theory but also offers a more adaptable and context-sensitive approach for understanding the complexities of global interactions.

In previous work, the authors defined a category SModF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SMod}_F$$\\end{document} of finite Galois modules decorated with local conditions for each global field F. In this paper, given an extension K/F of global fields, we define a restriction of scalars functor from SModK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SMod}_K$$\\end{document} to SModF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SMod}_F$$\\end{document} and show that it behaves well with respect to the Cassels–Tate pairing. We apply this work to study the class groups of global fields in the context of the Cohen–Lenstra heuristics.

In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First, we show that, over global fields, quadratic post-critically finite (PCF) polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes’ conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily PCF, and we combine our results with local class field theory to classify quadratic pairs over ${ {\mathbb{Q}}}$ with abelian dynamical Galois group, improving on recent results of Andrews and Petsche. On the other hand, we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence toward Jones’ finite index conjecture.

Let R be a field of rational functions of one variable over a field of constants R0. Dock Sang Rim (6) has proved that the global reciprocity law in exactly the usual sense holds whenever R0 is an absolutely algebraic quasi-fini te field of characteristic not equal to 0: this was known before only when R0 was a finite field. We shall give another proof of Rim's result by means of a noteworthy generalization of the usual global reciprocity law. Namely, let R0 be a finite field and let F be the set of all fields k contained in some fixed Ralg.clos. and of finite degree over R. The reciprocity law states that there exists a family {fk}, k ∈ F, of functions fk: Ck → G(kabel.clos./k) (where Ck is the idèle class group of k) enjoying certain properties such as the norm transfer law.

We present a new area law which is associated with the correlator of OPE blocks in higher dimensional conformal field theories (CFTs). The area law shows similar behaviour as black hole entropy or geometric entanglement entropy. It includes a leading term which is proportional to the area of the entanglement surface, and a logarithmic subleading term with degree q. We extract the UV cutoff independent coefficients and discuss various properties of the coefficients.

We develop Chevalley's Theorem and interesting implications. Revisiting Galois Cohomology we analyse the structure of the capitulation kernel. Moreover investigate the growth of ideal classes and consequences of a G-action on the class group. Finally we discuss capitulation in imaginary quadratic fields.

The class group of a number field K measures how far its ring of integers is from having unique factorization into irreducible elements. It is the quotient of the group of all fractional ideals of K by the subgroups of principal fractional ideals. It is well known from class field theory that the ideal class group is also the Galois group of the maximal unramified abelian extension of K.

This chapter continues global class field theory: reciprocity law, Brauer group, norm residue symbol.

Über die berechnung der schurschen indizes von Charakteren endlicher Gruppen

Higher dimensional self-similar spherical symmetric scalar field collapse and critical phenomena in black-hole formation

Transformers have recently gained significant attention in low-level vision tasks, particularly for remote sensing image super-resolution (RSISR). The vanilla vision transformer aims to establish long-range dependencies between image patches. However, its global receptive field leads to a quadratic increase in computational complexity with respect to spatial size, rendering it inefficient for addressing RSISR tasks that involve processing large-sized images. In an effort to mitigate computational costs, recent studies have explored the utilization of local attention mechanisms, inspired by convolutional neural networks (CNNs), focusing on interactions between patches within small windows. Nevertheless, these approaches are naturally influenced by smaller participating receptive fields, and the utilization of fixed window sizes hinders their ability to perceive multi-scale information, consequently limiting model performance. To address these challenges, we propose a hierarchical transformer model named the Multi-Scale and Global Representation Enhancement-based Transformer (MSGFormer). We propose an efficient attention mechanism, Dual Window-based Self-Attention (DWSA), combining distributed and concentrated attention to balance computational complexity and the receptive field range. Additionally, we incorporated the Multi-scale Depth-wise Convolution Attention (MDCA) module, which is effective in capturing multi-scale features through multi-branch convolution. Furthermore, we developed a new Tracing-Back Structure (TBS), offering tracing-back mechanisms for both proposed attention modules to enhance their feature representation capability. Extensive experiments demonstrate that MSGFormer outperforms state-of-the-art methods on multiple public RSISR datasets by up to 0.11–0.55 dB.