NMAG455
Course description:
Quadratic forms with integer coefficients played a crucial role in developing modern number theory, including many 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).
Notes from 2022/23 (in English, by Siu Hang Man)
Slides:
Lecture 1 (01/10): Introduction - Universal quadratic forms - Binary quadratic forms representing primes [See Cox - Ch. I, §1. A-B.]
Lecture 2 (08/10): Determinants - Lattices [See Dixon Ch. V, §50-51]
Lecture 3 (15/10): Reduced forms - Ternary quadratic forms - The three-square theorem [See Dixon Ch. IV, §25-27 (quadratic residues), §31-32, Ch. V, §52 (reduced forms)]
Lecture 4 (22/10): Ternary forms - Sums of squares [See Dixon Ch. V, §54-56] + Ramanujan's paper
Lecture 5 (29/10): Universal forms - Binary quadratic forms - Reduced forms [See Cox - Ch. I, §1. C, §2. A-B, Bhargava] + Conway's video
Lecture 6 (12/11): Class number - Genus theory [See Cox - Ch. I, §1. C]
Lecture 7 (19/11): Genus theory - Composition [See Cox - Ch. I, §2. C, §3. A] + Table of all genera of forms with I det I < 11
Lecture 8 (26/11): Composition and class groups [See Cox - Ch. I, §3. A]
Lecture 9 (03/12): Genus theory - Review of algebraic number theory [See Cox - Ch. I, §3. B, Ch. II, §5. A]
Lecture 10 (10/12): Review of algebraic number theory [See Cox - Ch. II, §5. A,B,C]
Lecture 11 (17/12): 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)