Arithmetic of Quadratic Forms Spring 2017

Organizers Menny Aka, Andreas Wieser Time Monday 13-15
(with exceptions)
Starting date 20.02.2017 Place HG D 3.2
Student meetings Fridays 16-18, HG G 26.3

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 mostly follow the beautiful monograph of Cassels "Rational quadratic forms" ([1]) 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.

Every student is assumed to give one talk in the first half and one talk in the second half of the semester, each of 45 minutes. A short explanation on how to prepare for a talk in this seminar can be found here. The speakers for week n will be determined at the end of the seminar in week n-1.

As a preparation for your talk, we offer you a non-mandatory meeting with us on the Friday before your talk (Fridays 16-18, HG G 26.3). We are of course also willing to answer questions by email.

Date Speakers Topics
20.02.17 Menny and Andreas Introduction to quadratic forms, organizational matters and generalities on quadratic forms over fields (Chapters 1 and 2.1 in [1])
27.02.17 Alain and Louis More generalities on quadratic forms over fields (isotropic subspaces, isometries and Witt's theorem) - Chapter 2 in [1]
06.03.17 Anna and Martin p-adic numbers - Chapter 3 in [1]
13.03.17 Vera and Nadir p-adic integers as an inverse limit, Hensel's Lemma and further properties of the Hilbert Symbol - Chapter 3 in [1], Chapter 2 in [2]
20.03.17 Noah and Elia Quadratic forms over local fields - Chapter 4 in [1]
27.03.17 Ajith and Felix Some geometry of numbers and the Hasse principle - Chapters 5 and 6 in [1]
03.04.17 Noah and Romain More on the Hasse principle - Chapter 6 in [1]
10.04.17 Martin and Elia An application of the Hasse principle and generalities on lattices over principal ideal domains (mostly Chapter 7 in [1])
26.04.17 Louis and Alain Integral p-adic forms (Chapter 8 in [1])
03.05.17 Vera and Anna Quadratic forms over the rational integers - Finiteness of the class number (Sections 1-3 of Chapter 9 in [1])
08.05.17 Romain and Felix Quadratic forms over the rational integers - Representating integers by integral rational quadratic forms (Sections 4-5 of Chapter 9 in [1])

Solving exercises is an important part of this seminar and a prerequisite for every attending student to obtain the credits. When you have solved an exercise, post it on this overleaf , so that we and other students can have a look at it. We expect every student to solve and post at least 4 exercises during the semester which should originate from at least 3 different chapters and which have yet not been solved by another student. What we mean by "chapter" is visible in overleaf . You may of course suggest other exercises of your interest.

Each week we will choose one solution to be presented to the class as a warm-up (5 minutes) at the beginning of the seminar. Naturally, you will be informed beforehand, if we want you to present one of your solved exercises.