##
**Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms. With an appendix by Dasheng Wei and Xu.**
*(English)*
Zbl 1190.11036

In this well-written and technically insightful article, the authors define an obstruction to the local-global principle for integer points on varieties, and apply it to the study of representation by integral quadratic forms.

The Brauer–Manin obstruction was put forward by [Yu. I. Manin, Actes Congr. Int. Math. 1970, No. 1, 401–411 (1971; Zbl 0239.14010)] as an explanation as to why certain varieties over number fields fail to satisfy the local-global principle: that is, they have points over every completion of the base field, but not over the base field itself. Manin’s construction has been applied fruitfully to the study of the arithmetic of many classes of varieties; a notable application from the point of view of this article is to homogeneous spaces under linear algebraic groups, for example by M. Borovoi [J. Reine Angew. Math. 473, 181–194 (1996; Zbl 0844.14020)].

In the present article, the authors take a scheme \(\mathcal X\) over a ring of \(S\)-integers of a number field, and define an obstruction to the local-global principle for integer points of \(\mathcal X\), using the Brauer group of the generic fibre. They study the Brauer pairings on homogeneous spaces under connected linear groups, and go on to apply these to obtain results about local-global principles for both rational and integer points on such homogeneous spaces. The results are obtained in two cases: where the stabilisers of points are connected (Section 3) and where the stabilisers are finite and commutative (Section 4).

Using these tools, the authors proceed to study the representation of quadratic forms by quadratic forms, which includes as a special case the representation of integers by quadratic forms. Suppose that we are given two nondegenerate quadratic forms \(f,g\) of ranks \(m,n\) respectively; then we seek linear forms \(l_1, \ldots, l_m\) such that \[ g(x_1, \ldots, x_n) = f( l_1(x_1, \ldots, x_n), \ldots, l_m(x_1, \ldots, x_n)). \] Solutions to a problem of this type are represented by points on a variety \(X\) which is a homogeneous space under a spin group; the stabiliser of a point may be a finite commutative group; another spin group; or a torus, depending on the relative ranks of the forms \(f\) and \(g\). In Section 5 this problem is studied over a field, using the results of the previous sections; in Section 6 the Brauer–Manin obstruction is applied to obtain criteria for the existence of integer solutions. In particular, computational recipes are given for testing solubility in integers.

Classically, questions about representation of quadratic forms by other quadratic forms have been phrased in terms of genera and spinor genera. In Section 7, the authors recall this language and proceed to show how the classical view relates very neatly to the Brauer–Manin obstruction. They also characterise the classical notion of spinor exception in the language of the Brauer–Manin obstruction.

The paper concludes with several well-chosen examples from the literature, interpreted using the Brauer–Manin obstruction. In particular, Section 9 investigates sums of three squares in an imaginary quadratic field.

An appendix by Xu and Dasheng Wei gives a result on sums of three squares in a cyclotomic field.

The Brauer–Manin obstruction was put forward by [Yu. I. Manin, Actes Congr. Int. Math. 1970, No. 1, 401–411 (1971; Zbl 0239.14010)] as an explanation as to why certain varieties over number fields fail to satisfy the local-global principle: that is, they have points over every completion of the base field, but not over the base field itself. Manin’s construction has been applied fruitfully to the study of the arithmetic of many classes of varieties; a notable application from the point of view of this article is to homogeneous spaces under linear algebraic groups, for example by M. Borovoi [J. Reine Angew. Math. 473, 181–194 (1996; Zbl 0844.14020)].

In the present article, the authors take a scheme \(\mathcal X\) over a ring of \(S\)-integers of a number field, and define an obstruction to the local-global principle for integer points of \(\mathcal X\), using the Brauer group of the generic fibre. They study the Brauer pairings on homogeneous spaces under connected linear groups, and go on to apply these to obtain results about local-global principles for both rational and integer points on such homogeneous spaces. The results are obtained in two cases: where the stabilisers of points are connected (Section 3) and where the stabilisers are finite and commutative (Section 4).

Using these tools, the authors proceed to study the representation of quadratic forms by quadratic forms, which includes as a special case the representation of integers by quadratic forms. Suppose that we are given two nondegenerate quadratic forms \(f,g\) of ranks \(m,n\) respectively; then we seek linear forms \(l_1, \ldots, l_m\) such that \[ g(x_1, \ldots, x_n) = f( l_1(x_1, \ldots, x_n), \ldots, l_m(x_1, \ldots, x_n)). \] Solutions to a problem of this type are represented by points on a variety \(X\) which is a homogeneous space under a spin group; the stabiliser of a point may be a finite commutative group; another spin group; or a torus, depending on the relative ranks of the forms \(f\) and \(g\). In Section 5 this problem is studied over a field, using the results of the previous sections; in Section 6 the Brauer–Manin obstruction is applied to obtain criteria for the existence of integer solutions. In particular, computational recipes are given for testing solubility in integers.

Classically, questions about representation of quadratic forms by other quadratic forms have been phrased in terms of genera and spinor genera. In Section 7, the authors recall this language and proceed to show how the classical view relates very neatly to the Brauer–Manin obstruction. They also characterise the classical notion of spinor exception in the language of the Brauer–Manin obstruction.

The paper concludes with several well-chosen examples from the literature, interpreted using the Brauer–Manin obstruction. In particular, Section 9 investigates sums of three squares in an imaginary quadratic field.

An appendix by Xu and Dasheng Wei gives a result on sums of three squares in a cyclotomic field.

Reviewer: Martin Bright (Warwick)

### MSC:

11G35 | Varieties over global fields |

11D85 | Representation problems |

11E12 | Quadratic forms over global rings and fields |

11E20 | General ternary and quaternary quadratic forms; forms of more than two variables |

11E57 | Classical groups |

11E72 | Galois cohomology of linear algebraic groups |

14G25 | Global ground fields in algebraic geometry |

14F22 | Brauer groups of schemes |

20G30 | Linear algebraic groups over global fields and their integers |

### Keywords:

integral quadratic forms; integral points; Hasse principle; Brauer-Manin obstruction; spinor exceptions; homogeneous spaces of linear algebraic groups; Galois cohomology
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\textit{J.-L. Colliot-Thélène} and \textit{F. Xu}, Compos. Math. 145, No. 2, 309--363 (2009; Zbl 1190.11036)

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