Introductory seminar about rational quadratic forms. P-adic numbers, Hasse's local to global principle and the finiteness of the genus will be discussed.
Learning objective
Quadratic forms and the numbers they represent have been of interest to mathematicians for a long time. For example, which integers can be expressed as a sum of two squares of integers? Or as a sum of three squares? Lagrange's four-squares theorem for instance states that any positive integer can be expressed as a sum of four squares. Such questions motivated the development of many aspects of algebraic number theory.
In this seminar we follow the beautiful monograph of Cassels "Rational quadratic forms" and will treat the fundamental results concerning quadratic forms over the integers and the rationals such as Hasse's local to global principle and finiteness of the genus.
Content
The seminar will mostly follow the book "Rational quadratic forms" by J.W.S. Cassels, particularly Chapters 1-9. Exercises in this book are an integral part of the seminar. Towards the end of the semester additional topics may be treated.
Lecture notes
Cassels, John William Scott. Rational quadratic forms. Vol. 13. Academic Pr, 1978.
Literature
Main reference: Cassels, John William Scott. Rational quadratic forms. Vol. 13. Academic Pr, 1978. Additional references:
Kitaoka, Yoshiyuki. Arithmetic of quadratic forms. Vol. 106. Cambridge University Press, 1999. Schulze-Pillot, Rainer. "Representation by integral quadratic forms - a survey." Contemporary Mathematics 344 (2004): 303-322.
Prerequisites / Notice
The student is assumed to have attended courses on linear algebra and algebra (as taught at ETH for instance). Previous knowledge on p-adic numbers is not assumed.
Performance assessment
Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course