Brauer Groups of Abelian Schemes
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No Transcendental Brauer-Manin Obstructions on Abelian Varieties
THERE ARE NO TRANSCENDENTAL BRAUER-MANIN OBSTRUCTIONS ON ABELIAN VARIETIES BRENDAN CREUTZ Abstract. Suppose X is a torsor under an abelian variety A over a number field. We show that any adelic point of X that is orthogonal to the algebraic Brauer group of X is orthogonal to the whole Brauer group of X. We also show that if there is a Brauer-Manin obstruction to the existence of rational points on X, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that A has finite Tate-Shafarevich group. Our results are unconditional. 1. Introduction Let X be a smooth projective and geometrically integral variety over a number field k. In order that X possesses a k-rational point it is necessary that X has points everywhere locally, i.e., that the set X(Ak) of adelic points on X is nonempty. The converse to this statement is called the Hasse principle, and it is known that this can fail. When X(k) is nonempty one can ask if weak approximation holds, i.e., if X(k) is dense in X(Ak) in the adelic topology. Manin [Man71] showed that the failure of the Hasse principle or weak approximation can, in many cases, be explained by a reciprocity law on X(Ak) imposed by the Brauer group, 2 Br X := Hét(X, Gm). Specifically, each element α ∈ Br X determines a continuous map, α∗ : X(Ak) → Q/Z, between the adelic and discrete topologies with the property that the subset X(k) ⊂ X(Ak) of rational points is mapped to 0. -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . -
Uniform Distribution in Subgroups of the Brauer Group of an Algebraic Number Field
Pacific Journal of Mathematics UNIFORM DISTRIBUTION IN SUBGROUPS OF THE BRAUER GROUP OF AN ALGEBRAIC NUMBER FIELD GARY R. GREENFIELD Vol. 107, No. 2 February 1983 PACIFIC JOURNAL OF MATHEMATICS Vol. 107, No. 2, 1983 UNIFORM DISTRIBUTION IN SUBGROUPS OF THE BRAUER GROUP OF AN ALGEBRAIC NUMBER FIELD GARY R. GREENFIELD We construct subgroups of the Brauer group of an algebraic number field whose member classes have Hasse invariants satisfying a rigid arithmetic structure — that of (relative) uniform distribution. After ob- taining existence and structure theorems for these subgroups, we focus on the problem of describing algebraic properties satisfied by the central simple algebras in these subgroups. Key results are that splitting fields are determined up to isomorphism, and there exists a distinguished subgroup of central automorphisms which can be extended. 1. Introduction. Let K be an algebraic number field, and let denote the class of the finite dimensional central simple X-algebra A in the Brauer group B(K) of K. The class [A] is determined arithmetically by its Hasse invariants at the primes of K. Algebraic properties of A often impose severe but interesting arithmetic properties on its invariants. As evidence we cite the important work of M. Benard and M. Schacher [2] concerning the invariants when [A] is in S(K) the Schur subgroup of K, and the surprising result of G. Janusz [4] obtained in considering the problem of when an automorphism of K extends to A. In this paper we offer a construction which gives rise to subgroups of B(K) whose member classes have invariants which possess a rigid arith- metic structure — that of uniform distribution — then search for corre- sponding algebraic properties. -
Non-Additive Ring and Module Theory IV the Brauer Group of a Symmetric
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 549 Brauer Groups Proceedings of the Conference Held at Evanston, October 11-15,1975 * 416 109 546 200 16 Edited by D. Zelinsky Springer-Verlag Berlin · Heidelberg · NewYork1976 Editor Daniel Zelinsky Northwestern University Department of Mathematics Evanston, II. 60091/USA Library of Congress Cataloging in Publication Data Main entry under title: Brauer groups. (Lecture notes in mathematics ; 5^9) "Sponsored by Northwestern University." Bi bli ography: ρ. Includes index. 1. Brauer group--Congresses. 2. Separable algebras—Congresses. I. Zelinsky, Daniel. II. Northwestern University, Evanston, 111. III. Series: Lecture notes in mathematics (Berlin) ; 5^9. QA3.L28 no. 5^9 [QA251.3] 510'.8s [512».2*0 76-kekie AMS Subject Classifications (1970): 13A20, 16A16, 18H20, 14C20, 14H99, 14L15, 18D10 ISBN 3-540-07989-0 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-07989-0 Springer-Verlag New York · Heidelberg · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin · Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. CONTENTS Lindsay N, Childs On Brauer groups of some normal local rings..... 1 P. -
More on Gauge Theory and Geometric Langlands
More On Gauge Theory And Geometric Langlands Edward Witten School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540 USA Abstract: The geometric Langlands correspondence was described some years ago in terms of S-duality of = 4 super Yang-Mills theory. Some additional matters relevant N to this story are described here. The main goal is to explain directly why an A-brane of a certain simple kind can be an eigenbrane for the action of ’t Hooft operators. To set the stage, we review some facts about Higgs bundles and the Hitchin fibration. We consider only the simplest examples, in which many technical questions can be avoided. arXiv:1506.04293v3 [hep-th] 28 Jul 2017 Contents 1 Introduction 1 2 Compactification And Hitchin’s Moduli Space 2 2.1 Preliminaries 2 2.2 Hitchin’s Equations 4 2.3 MH and the Cotangent Bundle 7 2.4 The Hitchin Fibration 7 2.5 Complete Integrability 8 2.6 The Spectral Curve 10 2.6.1 Basics 10 2.6.2 Abelianization 13 2.6.3 Which Line Bundles Appear? 14 2.6.4 Relation To K-Theory 16 2.6.5 The Unitary Group 17 2.6.6 The Group P SU(N) 18 2.6.7 Spectral Covers For Other Gauge Groups 20 2.7 The Distinguished Section 20 2.7.1 The Case Of SU(N) 20 2.7.2 Section Of The Hitchin Fibration For Any G 22 3 Dual Tori And Hitchin Fibrations 22 3.1 Examples 23 3.2 The Case Of Unitary Groups 25 3.3 Topological Viewpoint 27 3.3.1 Characterization of FSU(N) 29 3.4 The Symplectic Form 30 3.4.1 Comparison To Gauge Theory 33 4 ’t Hooft Operators And Hecke Modifications 34 4.1 Eigenbranes 34 4.2 The Electric -
The Brauer Group of the Moduli Stack of Elliptic Curves Over Algebraically Closed fields of Characteristic 2, J
The Brauer group of the moduli stack of elliptic curves Benjamin Antieau∗ and Lennart Meier† Abstract We compute the Brauer group of M1,1, the moduli stack of elliptic curves, over Spec Z, its localizations, finite fields of odd characteristic, and algebraically closed fields of characteristic not 2. The methods involved include the use of the parameter space of Legendre curves and the moduli stack M(2) of curves with full (naive) level 2 structure, the study of the Leray–Serre spectral sequence in ´etale cohomology and the Leray spectral sequence in fppf cohomology, the computation of the group cohomology of S3 in a certain integral representation, the classification of cubic Galois extensions of Q, the computation of Hilbert symbols in the ramified case for the primes 2 and 3, and finding p-adic elliptic curves with specified properties. Key Words. Brauer groups, moduli of elliptic curves, level structures, Hilbert sym- bols. Mathematics Subject Classification 2010. Primary: 14F22, 14H52, 14K10. Secondary: 11G05, 11G07. Contents 1 Introduction 2 2 Brauer groups, cyclic algebras, and ramification 4 3 The low-dimensional Gm-cohomology of BCm 10 4 A presentation of the moduli stack of elliptic curves 13 arXiv:1608.00851v3 [math.AG] 5 May 2020 5 Beginning of the computation 17 6 The p-primary torsion in Br(MZ[ 1 ]) for primes p 5 21 2 ≥ 7 The 3-primary torsion in Br(M ) 22 Z 1 [ 6 ] 8 The ramification of the 3-torsion 24 9 The 2-primary torsion in Br(M 1 ) 26 Z[ 2 ] 10 The Brauer group of M 30 11 The Brauer group of M over Fq with q odd 31 ∗Benjamin Antieau was supported by NSF Grants DMS-1461847 and DMS-1552766. -
Arxiv:1711.06456V4 [Math.AG] 1 Dec 2018
PURITY FOR THE BRAUER GROUP KĘSTUTIS ČESNAVIČIUS Abstract. A purity conjecture due to Grothendieck and Auslander–Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension ě 2. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic p0,pq. We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover. 1. The purity conjecture of Grothendieck and Auslander–Goldman .............. 1 Acknowledgements ................................... ................................ 3 2. Passage to a finite flat cover ................................................... .... 3 3. Passage to the completion ................................................... ....... 6 4. The p-primary Brauer group in the perfectoid case .............................. 7 5. Passage to perfect or perfectoid towers ........................................... 11 6. Global conclusions ................................................... ............... 13 Appendix A. Fields of dimension ď 1 ................................................ 15 References ................................................... ........................... 16 1. The purity conjecture of Grothendieck and Auslander–Goldman Grothendieck predicted in [Gro68b, §6] that the cohomological Brauer group of a regular scheme X is insensitive to removing a closed subscheme Z Ă X of codimension ě 2. This purity conjecture is known in many cases (as we discuss in detail below), for instance, for cohomology classes of order 2 invertible on X, and its codimension requirement is necessary: the Brauer group of AC does not agree with that of the complement of the coordinate axes (see [DF84, Rem. 3]). In this paper, we finish the remaining cases, that is, we complete the proof of the following theorem. -
Brauer Groups and Galois Cohomology of Function Fields Of
Brauer groups and Galois cohomology of function fields of varieties Jason Michael Starr Department of Mathematics, Stony Brook University, Stony Brook, NY 11794 E-mail address: [email protected] Contents 1. Acknowledgments 5 2. Introduction 7 Chapter 1. Brauer groups and Galois cohomology 9 1. Abelian Galois cohomology 9 2. Non-Abelian Galois cohomology and the long exact sequence 13 3. Galois cohomology of smooth group schemes 22 4. The Brauer group 29 5. The universal cover sequence 34 Chapter 2. The Chevalley-Warning and Tsen-Lang theorems 37 1. The Chevalley-Warning Theorem 37 2. The Tsen-Lang Theorem 39 3. Applications to Brauer groups 43 Chapter 3. Rationally connected fibrations over curves 47 1. Rationally connected varieties 47 2. Outline of the proof 51 3. Hilbert schemes and smoothing combs 54 4. Ramification issues 63 5. Existence of log deformations 68 6. Completion of the proof 70 7. Corollaries 72 Chapter 4. The Period-Index theorem of de Jong 75 1. Statement of the theorem 75 2. Abel maps over curves and sections over surfaces 78 3. Rational simple connectedness hypotheses 79 4. Rational connectedness of the Abel map 81 5. Rational simply connected fibrations over a surface 82 6. Discriminant avoidance 84 7. Proof of the main theorem for Grassmann bundles 86 Chapter 5. Rational simple connectedness and Serre’s “Conjecture II” 89 1. Generalized Grassmannians are rationally simply connected 89 2. Statement of the theorem 90 3. Reductions of structure group 90 Bibliography 93 3 4 1. Acknowledgments Chapters 2 and 3 notes are largely adapted from notes for a similar lecture series presented at the Clay Mathematics Institute Summer School in G¨ottingen, Germany in Summer 2006. -
Elliptic Spectra, the Witten Genus and the Theorem of the Cube
ELLIPTIC SPECTRA, THE WITTEN GENUS AND THE THEOREM OF THE CUBE M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND Contents 1. Introduction 2 1.1. Outline of the paper 6 2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7 2.2. Constructions of elliptic spectra 9 2.3. The complex-orientable homology of BUh2ki for k ≤ 3 10 2.4. The complex-orientable homology of MUh2ki for k ≤ 3 14 2.5. The σ–orientation of an elliptic spectrum 18 2.6. The Tate curve 20 2.7. The elliptic spectrum KTate and its σ-orientation 23 2.8. Modularity 25 k 3. Calculation of C (Gb a, Gm) 26 3.1. The cases k = 0 and k = 1 26 3.2. The strategy for k = 2 and k = 3 26 3.3. The case k = 2 27 3.4. The case k = 3: statement of results 28 3.5. Additive cocycles 30 3.6. Multiplicative cocycles 32 2 b 3 b 3.7. The Weil pairing: cokernel of δ× :C (Ga, Gm) →C (Ga, Gm) 35 1 b 2 b 3.8. The map δ× :C (Ga, Gm) →C (Ga, Gm) 39 3.9. Rational multiplicative cocycles 40 4. Topological calculations 40 4.1. Ordinary cohomology 41 4.2. The isomorphism for BUh0i and BUh2i 41 4.3. The isomorphism for rational homology and all k 42 4.4. The ordinary homology of BSU 42 4.5. The ordinary homology of BUh6i 43 4.6. BSU and BUh6i for general E 46 Appendix A. -
Brauer Groups of Linear Algebraic Groups
Volume 71, Number 2, September 1978 BRAUERGROUPS OF LINEARALGEBRAIC GROUPS WITH CHARACTERS ANDY R. MAGID Abstract. Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)<n) where C is finite and n = d(d - l)/2, with d the Z-Ta.nk of the character group of G. In particular, a linear torus of dimension d has Brauer group (Q/Z)(n) with n as above. In [6], B. Iversen calculated the Brauer group of a connected, characterless, linear algebraic group over an algebraically closed field of characteristic zero: the Brauer group is finite-in fact, it is the Schur multiplier of the fundamental group of the algebraic group [6, Theorem 4.1, p. 299]. In this note, we extend these calculations to an arbitrary connected linear group in characteristic zero. The main result is the determination of the Brauer group of a ¿/-dimen- sional affine algebraic torus, which is shown to be (Q/Z)(n) where n = d(d — l)/2. (This result is noted in [6, 4.8, p. 301] when d = 2.) We then show that if G is a connected linear algebraic group whose character group has Z-rank d, then the Brauer group of G is C X (Q/Z)<n) where n is as above and C is finite. We adopt the following notational conventions: 7ms an algebraically closed field of characteristic zero, and T = (F*)id) is a ^-dimensional affine torus over F. -
Cohomological Interpretation of Brauer Groups 20172252 Seewoo Lee
Cohomological interpretation of Brauer Groups 20172252 Seewoo Lee Abstract In this note, we introduce a centeral simple algebra and a Brauer group, which classifies central division algebra over a given field. We prove that there exists a canonical isomorphism between a relative Brauer group Br(L=k) and the second group cohomology H2(Gal(L=k);L×), which shows that the cohomology of group can be used to classify central division algebras over a given field. We also present some examples and computations. 1 Introduction Let k be a field. A central simple algebra A over k is a finite dimensional associative algebra which is simple and center is k. For example, the quoternions H is a 4-dimensional central simple algebra over R. Our basic question is the following: Classify the central simple algebras over a given field k. This seems very hard problem at first glance. However, we will show that the Artin-Wedderburn theorem claims that every central simple algebra over k is isomorphic to a matrix algebra over some central division algebra over k which is determined unique up to isomorphism. Hence the theorem reduces the problem as Classify the central division algebras over a given field k. However, this isn't an easy problem. To do this, we will define the Brauer group Br(k) of a given field, which is a group that classifies central division algebras over k. More precisely, each elements of Br(k) corresponds to the isomorphism class of the central division algebras over k. Computation of the Brauer group can be done by the computation of the second cohomology group, by the following main theorem we will prove: Br(L=k) ' H2(Gal(L=k);L×): Using this, we will compute Brauer groups over various fields such as R; Fp; Qp; Q. -
Central Simple Algebras and the Brauer Group
Central Simple Algebras and the Brauer group XVIII Latin American Algebra Colloquium Eduardo Tengan (ICMC-USP) Copyright c 2009 E. Tengan Permission is granted to make and distribute verbatim copies of this document provided the copyright notice is preserved on all copies. The author was supported by FAPESP grant 2008/57214-4. Chapter 1 CentralSimpleAlgebrasandthe Brauergroup 1 Some conventions Let A be a ring. We denote by Mn(A) the ring of n n matrices with entries in A, and by GLn(A) its group of units. We also write Z(A) for the centre of A×, and Aop for the opposite ring, which is the ring df with the same underlying set and addition as A, but with the opposite multiplication: a Aop b = b A a op ≈ op × × for a,b A . For instance, we have an isomorphism Mn(R) Mn(R) for any commutative ring R, given by∈M M T where M T denotes the transpose of M. → 7→ A ring A is simple if it has no non-trivial two-sided ideal (that is, an ideal different from (0) or A). A left A-module M is simple or irreducible if has no non-trivial left submodules. For instance, for any n field K, Mn(K) is a simple ring and K (with the usual action) is an irreducible left Mn(K)-module. If K is a field, a K-algebra D is called a division algebra over K if D is a skew field (that is, a “field” with a possibly non-commutative multiplication) which is finite dimensional over K and such that Z(D)= K.