Quadratic forms and class fields I

NMAG455

Autumn semester 2020/21 (previous years)

Course description:

Quadratic forms with integer coefficients played a crucial role in the development of the modern number theory, including many of the tools in algebraic number theory, class field theory and modular forms. The lectures aim to explain the basics of the arithmetic theory of quadratic forms, especially questions concerning the representation of prime numbers in the form of x^2+ny^2.

Notes:

Scanned notes (in Czech, by Vita Kala).

Slides:

Lecture 1 (30/09): Slides + Video (Introduction - Universal quadratic forms - Binary forms representing primes) [See Cox - Ch. I, §1. A-B.]

Lecture 2 (07/10): Slides + Video (Determinants and discriminants - Ternary forms - Lattices) [See Dixon Ch. V, §50-51]

Lecture 3 (14/10): Slides + Video (Reduced binary and ternary forms - Sum of squares) [See Dixon Ch. IV, §25-27 (quadratic residues), §31-32, Ch. V, §52 (reduced forms)]

Lecture 4 (21/10): Slides + Video (Ternary forms - Sums of squares - Universal forms) [See Dixon Ch. V, §54-56, Bhargava] + Ramanujan's paper + Conway's video

Lecture 5 (04/11): Slides + Video (Quaternions - Sum of four squares)

Lecture 6 (11/11): Slides + Video (Binary quadratic forms - Reduced forms - Class number) [See Cox - Ch. I, §1. C, §2. A-B]

Lecture 7 (18/11): Slides + Video (Genus theory - Composition) [See Cox - Ch. I, §2. C, §3. A] + Table of all genera of forms with I det I < 11

Lecture 8 (25/11): Slides + Video (Composition and class groups) [See Cox - Ch. I, §3. A]

Lecture 9 (02/12): Slides + Video (Genus theory) [See Cox - Ch. I, §3. B]

Lecture 10 (09/12): Slides + Video (Review of algebraic number theory) [See Cox - Ch. II, §5. A,B,C]

Lecture 11 (16/12): Slides + Video (Primes of the form p=x^2+ny^2 - Orders and quadratic forms) [See Cox - Ch. II, §5. D, §7. B]

Recommended literature:

Modern elementary theory of numbers - L. E. Dickson

Primes of the forms x^2+ny^2 - D. A. Cox

Additional literature:

On the Conway–Schneeberger fifteen theorem - M. Bhargava (Universal forms)

Arithmetic of quadratic forms - W. K. Chan (A general reference, more advanced material)

On the Classification of Integral Quadratic Forms - J.H. Conway and N. J. A. Sloane (§1-3 and 10. Reduced forms)

Universal Quadratic Forms and the Fifteen Theorem - J. H. Conway (Universal forms)

On the expression of a number in the form ax² + by² + cz² + du - S. Ramanujan (Universal forms)

Lecture Notes on Quadratic Forms and their Arithmetic - R. Schulze-Pillot (A general reference, more advanced material)