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Lecture

Quadratic Lattices and Genus Classes

In course

MATH-603: Subconvexity, Periods and Equidistribution

This course is a modern exposition of "Duke's Theorems" which describe the distribution of representations of large integers by a fixed ternary quadratic form. It will be the occasion to introduce the students to the adelic language, the theory of automorp ...

Description

This lecture covers the integral Hasse principle for locally representable integers by quadratic lattices, the concept of locally isometric lattices, and the Hasse-Minkowski theorem. It also discusses the Hermite-Minkowski theorem, equidistribution in the genus, and the local-global principle for lattices. The lecture concludes with the ring of adèles of Q, the ring of finite adèles, and the possible completions of Q indexed by the King of Adeles.

Instructor

Philippe Michel

Ph. Michel's main research interest lie in the field of analytic number theory and range over a variety of techniques and methods which include: arithmetic geometry, exponential sums, sieve methods, automorphic forms and allied representations, L-functions and more recently ergodic theory. Ph. Michel is a former student of ENS Cachan and obtained his PhD in Universté Paris XI in 1995 under the guidance of E. Fouvry. From 1995 to 1998 he was maître de conférence at Universté Paris XI and full professor at Université Montpellier II until 2008 then when he joined EPFL. Ph. Michel was awarded the Peccot-Vimont prize, has been member of the Institut Universitaire de France and wa invited speaker at the 2006 International Congress of Mathematician.

Official source

Related concepts (31)

Adele ring

In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring. An adele derives from a particular kind of idele. "Idele" derives from the French "idèle" and was coined by the French mathematician Claude Chevalley. The word stands for 'ideal element' (abbreviated: id.el.

Hasse principle

In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.

Mathematics

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them.

Brauer group

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer. The Brauer group arose out of attempts to classify division algebras over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles.

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Related lectures (22)

Integral Quadratic Lattices: Properties and Adeles

Covers non-degenerate integral quadratic lattices and adèles, including their properties and definitions.

Quadratic Lattices: Properties and Equidistribution

Covers non-degenerate quadratic lattices, local representability, equidistribution, and the Siegel theorem.

Integral Representations: Quadratic Lattices and Hasse Principle

Explores integral representations, quadratic lattices, and the Hasse Principle.

Topology of Adeles

Covers the topology of Adeles and their relationship with quadratic forms, polynomial varieties, and finiteness properties.

Adelic Topology: Properties and Approximation