Quadratic forms, algebraic groups and number theory

N. M. Glazunov

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 77-89.

Résumé

The aim of the article is an overview of some important results in the theory of quadratic forms, and algebraic groups, and which had an impact on the development of the theory of numbers. The article focuses on selected tasks and is not exhaustive. A mathematical structures, methods and results, including the new ones, related in some extent with research of V.P. Platonov. The content of the article is following. In the introduction drawn attention to the classic researches of Korkin, Zolotarev and Voronoi on the theory of extreme forms and recall the relevant definitions. In section 2 "Quadratic forms and lattices" presented the necessary definitions, the results of the lattices and quadratic forms over the field of real numbers and over the ring of rational integers. Section 3 "Algebraic groups" contains a representation of the class of lattices in a real space as factors of algebraic groups, as well as the version of Mahler's compactness criterion of such factors. Bringing the results of the compactness of factors of orthogonal groups of quadratic forms which do not represent zero rationally, and the definitions and concepts related to the quaternion algebras over rational numbers. These results explicitly or implicitly are used in the works of V. P. Platonov and in sections 4 and 5. Section 4 " Heegner points and their generalizations" provides an overview of new research in the direction of finding Heegner points and their generalizations. Section 5 summarizes some new research and results on the Hasse principle for algebraic groups. For the reading of the article may be a useful another article which has published by the author in the Chebyshevsky sbornik, vol. 16, no. 3, in 2015. I am deeply grateful to N. M. Dobrovolskii for help and support under the preparation of the article for publication. Bibliography: 31 titles.

Détail

Keywords: positive definite quadratic form; finite-dimensional associative division algebra over rationals; Hasse principle; rigidity; Heegner point.

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