Syllabus
Basic notions: equivalence of quadratic forms, determinant, associated matrix and lattice, reduction of forms
Ternary forms, 3- and 4-square theorems, universal diagonal forms, 15 theorem
Binary forms: composition and form class group, genus theory
Isomorphism of ideal and form class groups
Hilbert class field and primes of the form x^2+ny^2
Annotation
Quadratic forms with integral coefficients form a central part of number theory - for example, the study of primes represented by the form x^2+ny^2 gradually led to the development of many key tools in algebraic number theory, ranging from the study of number fields to the theory of class fields and modular forms. The goal of the course is to explain the basics of the arithmetic theory of quadratic forms, in particular with focus on the question of representability of integers including applications of class field theory.
The course may not be taught every academic year.