scholarly journals Global Jacquet-Langlands Correspondence for Division Algebras in Characteristic $p$

2016 ◽  
pp. rnw094 ◽  
Author(s):  
Alexandru Ioan Badulescu ◽  
Philippe Roche
Keyword(s):  
Division Algebras ◽  
Characteristic P ◽  
Langlands Correspondence

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

A torsion Jacquet-Langlands correspondence

Astérisque ◽  
2019 ◽  
Vol 409 ◽  
pp. 1-226 ◽  
Author(s):  
Frank CALEGARI ◽  
Akshay VENKATESH
Keyword(s):  
Langlands Correspondence

scholarly journals Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

2021 ◽  
Vol 9 ◽  
Author(s):  
Benjamin Antieau ◽  
Bhargav Bhatt ◽  
Akhil Mathew
Keyword(s):  
Spectral Sequence ◽  
Spectral Sequences ◽  
Characteristic P ◽  
De Rham

Abstract We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

Effect of oxygen vacancies and crystal symmetry on piezocatalytic properties of Bi2WO6 ferroelectric nanosheets for wastewater decontamination

2021 ◽  
Author(s):  
Zihan Kang ◽  
Enzhu Lin ◽  
Ni Qin ◽  
Jiang Wu ◽  
Baowei Yuan ◽  
...  
Keyword(s):  
Organic Pollutants ◽  
Oxygen Vacancies ◽  
Mechanical Energy ◽  
Crystal Symmetry ◽  
Characteristic P ◽  
Novel Technique ◽  
Effect Of Oxygen

Piezocatalysis emerged as a novel technique to make use of mechanical energy in dealing with organic pollutants in wastewater. In this work, the ferroelectric Bi2WO6 (BWO) nanosheets with a characteristic...

scholarly journals Trace dynamics and division algebras: towards quantum gravity and unification

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Tejinder P. Singh
Keyword(s):  
Lie Group ◽  
Planck Scale ◽  
The Self ◽  
Division Algebras ◽  
Quantum Field ◽  
Octonion Algebra ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Massless Spin ◽  
Correct Understanding

AbstractWe have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang–Mills fields, and fermions. Dynamical variables are described by odd-grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper, we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G2, replace space-time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1)em, describes the electrocolour symmetry, as shown earlier by Furey. We predict a new massless spin one boson (the ‘Lorentz’ boson) which should be looked for in experiments. Our Lagrangian correctly describes three fermion generations, through three copies of the group G2, embedded in the exceptional Lie group F4. This is the unification group for the four fundamental interactions, and it also happens to be the automorphism group of the exceptional Jordan algebra. Gravitation is shown to be an emergent classical phenomenon. Although at the Planck scale, there is present a quantised version of the Lorentz symmetry, mediated by the Lorentz boson, we argue that at sub-Planck scales, the self-adjoint part of the octonionic trace dynamics bears a relationship with string theory in 11 dimensions.

scholarly journals On division algebras having the same maximal subfields

2010 ◽  
Vol 132 (3-4) ◽  
pp. 273-293 ◽  
Author(s):  
Andrei S. Rapinchuk ◽  
Igor A. Rapinchuk
Keyword(s):  
Division Algebras
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