A relation between the Brauer group and the Tate-Shafarevich group
type: Item
instance of: scholarly article
title: A relation between the Brauer group and the Tate-Shafarevich group
zbMATH Open document ID: 1302.11039
author: Chuangxun Cheng
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2013-03-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1356012912
review text: Let \(K\) be a number field or a function field, and \(C \) a smooth geometrically connected curve over \(K\). For each place \(v\) of \(K\) let \(\delta_v\) denote the local index of \(C_{K_v}\), and \(\delta\) the index of \(C\). Let \(A\) denote the Jacobian variety of \(C\), \(\text Ш(A)\) the Tate-Shafarevich group of \(A\), and \(\mathrm{Br}(C)\) the Brauer group of \(C\). Define \[ \mathrm{Br}(C)':=\mathrm {ker}\big[ \mathrm{Br}(C) \to \oplus_v \mathrm{Br}(C_{K_v})\big]. \] Motivated by the works of Grothendieck and of Artin, \textit{J. S. Milne} [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 735--743 (1981; Zbl 0503.14010)] conditionally extended Artin's result and refined Grothendieck's exact sequence as follows: suppose that \(\delta_v=1\) for all \(v\). If \(\text Ш(A)\) has no nonzero infinitely divisible elements, then we have an exact sequence \[ 0 \to \mathrm{Br}(C)' \to \text Ш(A)/T_1 \to T_2 \to 0 \] where \(\# T_1 = \delta\) and \(\#T_2 = \delta/p^r\) where \(p\) is the characteristic of \(K\) and \(p^r=1\) if \(K\) is a number field. If one of \(\text Ш(A)\) or \(\mathrm{Br}(C)'\) is finite, then so is the other, and \(\delta^2\cdot\#\mathrm{Br}(C)' = p^r\cdot\#\text Ш(A)\). Motivated by \textit{P. L. Clark} [J. Reine Angew. Math. 594, 201--206 (2006; Zbl 1097.14024)], here the author considers a number field \(K\), a place \(w\) and, a curve \(C\) of genus \(1\) that corresponds to an element in \(\ker\big[ \mathsf{H}^1( K,E) \to \oplus_{v\neq w}\mathsf{H}^1(K_v,E)\big]\) where \(E\) is the Jacobian of \(C\), which is an elliptic curve. Hence, \(\delta_v=1\) for all \(v\) except for \(v=w\). Let \(d\) and \(d_w\) be the indices of \(C\) and \(C_{K_w}\). The author obtains an exact sequence for \(C\) same as Milne's, and proves \[ (d/d_w)^2\cdot\#\mathrm{Br}(C)' = \#\text Ш(E) \] provided that one of \(\text Ш(E)\) or \(\mathrm{Br}(C)'\) is finite. In fact, \textit{C. D. Gonzalez-Avilez} obtains a more general result in [J. Math. Sci., Tokyo 10, No. 2,…
reviewed by: Sungkon Chang
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 6148880
zbMATH Keywords: Brauer group
zbMATH Keywords: Tate-Shafarevich group
MaRDI profile type: Publication
OpenAlex ID: W2045952456
cites work: There are genus one curves of every index over every number field
cites work: Q4432888
cites work: The Period-Index Problem for Elliptic Curves
cites work: Duality theorems for curves over p-adic fields
cites work: Q3969052
cites work: Q3710637
DOI: 10.7169/FACM/2012.47.2.3
description: scientific article; zbMATH DE number 6148880