On the Brauer group of a surface
type: Item
instance of: scholarly article
title: On the Brauer group of a surface
zbMATH Open document ID: 1077.14023
author: Qing Liu
author: Michel Raynaud
author: Dino Lorenzini
published in: Inventiones Mathematicae
publication date: 2005-04-14T00:00:00Z
review text: The authors establish the following theorem in full generality: Let \(X\) be a smooth proper surfaces over a finite field \(k\) of characteristic \(p\). Suppose that the Brauer group of \(X\) has finite \(l\)-part for some prime number \(l\). Then the order of the Brauer group is a square. Here the assumption on the \(l\)-part is needed to ensure that the Brauer group is finite. The story of this remarkable theorem starts with results of \textit{J. Tate} [in: Dix Exposés Cohomologie Schémas, Adv. Stud. Pure Math. 3, 189--214 (1968; Zbl 0199.55604)], who constructed an antisymmetric pairing on the prime-to-\(p\) part of \(\text{ Br}(X)\), whose radical equals the subgroup of divisible elements. If the entire Brauer group is finite, and furthermore carries an alternating nondegenerate pairing, it would follow that the order of the Brauer group must be a square. Instead of looking for such a pairing, the authors follow a completely different line of reasoning. The basic steps are as follows: First, they modify the surface \(X\) so that there is a fibration \(X\rightarrow\mathbb P^1\) with smooth generic fiber. Care must be taken because we are over a finite field, but things work well due to results of \textit{O. Gabber} [Geom. Funct. Anal. 11, 1192--1200 (2001; Zbl 1072.14513)]. Next, they examine the generic fiber \(X_K/K\) and its Jacobian \(A_K/K\), where \(K\) is the function field of the projective line. Using results \textit{K. Kato} and \textit{F. Trihan} [Invent. Math. 153, 537--592 (2003; Zbl 1046.11047)], they observe, as Saper pointed out, that the conjecture of Birch-Swinnerton-Dyer on the \(L\)-function of the abelian variety \(A_K/K\) is equivalent to the conjecture of Artin-Tate on the Brauer group of the surface \(X\). The equvialence of these two conjectures was itself a conjecture, sometimes called conjecture d) of Artin-Tate. In an earlier paper [Invent. Math. 157, 455--518 (2004; Zbl 1060.14037)], the authors showed, assuming that the Brauer group …
reviewed by: Stefan Schröer
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 2156035
zbMATH Keywords: Brauer group
zbMATH Keywords: Artin-Tate Conjecture
zbMATH Keywords: Birch-Swinnerton-Dyer Conjecture
MaRDI profile type: Publication
full work available at URL: s00222-004-0403-2
OpenAlex ID: W1984329303
cites work: On space filling curves and Albanese varieties.
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Q4143433
cites work: Complexe de de\thinspace Rham-Witt et cohomologie cristalline
cites work: On the conjectures of Birch and Swinnerton-Dyer in characteristic \(p>0\)
cites work: Q4530238
cites work: Néron models, Lie algebras, and reduction of curves of genus one
cites work: The Brauer group of a rational surface
cites work: On a conjecture of Artin and Tate
cites work: Duality in the flat cohomology of a surface
cites work: Bertini theorems over finite fields
cites work: The Cassels-Tate pairing on polarized Abelian varieties
cites work: Calculation of Manin’s invariant for Del Pezzo surfaces
cites work: The bilinear form of the Brauer group of a surface
DOI: 10.1007/S00222-004-0403-2
Recommended article: COMPARING THE BRAUER GROUP TO THE TATE–SHAFAREVICH GROUP
Recommended article: TATE’S CONJECTURE AND THE TATE–SHAFAREVICH GROUP OVER GLOBAL FUNCTION FIELDS
Recommended article: On the Brauer group of an arithmetic scheme
Recommended article: The bilinear form of the Brauer group of a surface
Recommended article: The conjectures of Artin-Tate and Birch-Swinnerton-Dyer
Recommended article: Q3682595
Recommended article: On the conjectures of Birch and Swinnerton-Dyer in characteristic \(p>0\)
Recommended article: A remark on uniform boundedness for Brauer groups
Recommended article: Uniform boundedness for Brauer groups of forms in positive characteristic
Recommended article: Étale Steenrod operations and the Artin–Tate pairing
description: scientific article; zbMATH DE number 2156035