Brauer groups, Tamagawa measures, and rational points on algebraic varieties.

*(English)*Zbl 1308.14025
Mathematical Surveys and Monographs 198. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1882-3/hbk). viii, 267 p. (2014).

This book is concerned with the existence and distribution of rational points on algebraic varieties. Thus it focuses in particular on the Hasse principle and the Brauer–Manin obstruction, and on the Manin conjecture.

The book is divided into 3 parts. Part A, on heights, describes the notion of height, and introduces the conjectures of Lang, of Batyrev and Manin, and of Manin and Peyre. There is a full account of the different factors in the Peyre constant, and a discussion of some of the proven cases, and the methods used for them.

Part B concerns the Brauer group. Here one learns firstly the general theory of the Brauer group. This is then applied to the Brauer–Manin obstruction, developing the general theory before going on to apply it to a range of special cubic surfaces. In particular the case of diagonal cubic surfaces is presented in some detail, describing some, but not all, of the results of J.-L. Colliot-Thélène et al. [Lect. Notes Math. 1290, 1–108 (1987; Zbl 0639.14018)].

The third part of the book, entitled “Numerical experiments” describes three different numerical questions. The first is the search for rational points on the quartic surface \(x^4+2y^4=z^4+4w^4\). Swinnerton-Dyer had asked whether there were any rational points other than the obvious ones with \(y=w=0\), and the first chapter of this section describes the way one conducts an efficient search for such points, leading eventually to the solution \[ 1484801^4+2.1203120^4=1169407^4+4.1157520^4. \] The second chapter describes an investigation into Manin’s conjecture for 3-folds of the shape \(ax^3=by^3+z^3+v^3+w^3\) and \(ax^4=by^4+z^4+v^4+w^4\), with \(1\leq a,b\leq 100\). Again there is a search procedure, locating points of height up to 5000 in the cubic case, and 100000 in the quartic case. One also has to compute the Peyre constants, which in particular involves all the local densities. In addition one must identify the various accumulating subvarieties. All this is described in the chapter, giving a comprehensive practical account of everything involved in the Manin–Peyre conjecture. The final chapter in the book concerns the height of the smallest rational point on a diagonal cubic surface \(X\). In particular one may ask to what extent the smallest height of such a point, \(m(X)\) say, can be bounded in terms of the Peyre constant \(\tau(X)\). Indeed one could conjecture that \(m(X)=O_{\alpha}(\tau(X)^{-\alpha})\) for any \(\alpha>0\). A numerical investigation of over 800000 surfaces is described, and again the calculation of \(\tau(X)\) plays a major role in the computations.

Overall this book delivers a thorough grounding in all aspects of the Hasse principle and the Brauer–Manin obstruction, and the Manin conjecture. It is to be recommended for serious study by research students and more senior workers, but the wealth of examples, and of different angles on the material also make it suitable for dabbling by the more general reader.

The book is divided into 3 parts. Part A, on heights, describes the notion of height, and introduces the conjectures of Lang, of Batyrev and Manin, and of Manin and Peyre. There is a full account of the different factors in the Peyre constant, and a discussion of some of the proven cases, and the methods used for them.

Part B concerns the Brauer group. Here one learns firstly the general theory of the Brauer group. This is then applied to the Brauer–Manin obstruction, developing the general theory before going on to apply it to a range of special cubic surfaces. In particular the case of diagonal cubic surfaces is presented in some detail, describing some, but not all, of the results of J.-L. Colliot-Thélène et al. [Lect. Notes Math. 1290, 1–108 (1987; Zbl 0639.14018)].

The third part of the book, entitled “Numerical experiments” describes three different numerical questions. The first is the search for rational points on the quartic surface \(x^4+2y^4=z^4+4w^4\). Swinnerton-Dyer had asked whether there were any rational points other than the obvious ones with \(y=w=0\), and the first chapter of this section describes the way one conducts an efficient search for such points, leading eventually to the solution \[ 1484801^4+2.1203120^4=1169407^4+4.1157520^4. \] The second chapter describes an investigation into Manin’s conjecture for 3-folds of the shape \(ax^3=by^3+z^3+v^3+w^3\) and \(ax^4=by^4+z^4+v^4+w^4\), with \(1\leq a,b\leq 100\). Again there is a search procedure, locating points of height up to 5000 in the cubic case, and 100000 in the quartic case. One also has to compute the Peyre constants, which in particular involves all the local densities. In addition one must identify the various accumulating subvarieties. All this is described in the chapter, giving a comprehensive practical account of everything involved in the Manin–Peyre conjecture. The final chapter in the book concerns the height of the smallest rational point on a diagonal cubic surface \(X\). In particular one may ask to what extent the smallest height of such a point, \(m(X)\) say, can be bounded in terms of the Peyre constant \(\tau(X)\). Indeed one could conjecture that \(m(X)=O_{\alpha}(\tau(X)^{-\alpha})\) for any \(\alpha>0\). A numerical investigation of over 800000 surfaces is described, and again the calculation of \(\tau(X)\) plays a major role in the computations.

Overall this book delivers a thorough grounding in all aspects of the Hasse principle and the Brauer–Manin obstruction, and the Manin conjecture. It is to be recommended for serious study by research students and more senior workers, but the wealth of examples, and of different angles on the material also make it suitable for dabbling by the more general reader.

Reviewer: D. R. Heath-Brown (Oxford)

##### MSC:

14G05 | Rational points |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11D45 | Counting solutions of Diophantine equations |

11G50 | Heights |