Title of the course: Quadratic forms
Instructor: Dr. Katharina Hübner
Institution: Universität Heidelberg
Dates: 23-29 September 2019
Prerequisites: Algebra, p-adic numbers
Level: Advanced undergraduate
Abstract: Suppose we want to solve a Diophantine equation, i.e. we are looking for rational solutions of a polynomial equation with integer coefficients. A standard approach is to first check whether there are solutions in Qp for varying p and in R, which is a much easier task. Obviously this is a necessary condition as every solution in Q is also a solution in Qp and in R.
The converse is not true in general: There are Diophantine equations that have solutions in Qp for all p and in R but not in Q. Although these are rather the exception, it is hard to prove when this is the case. If the polynomial has degree 2, however, it has a solution in Q if and only if it has some solution in each of the Qp’s and in R. This is the Hasse-Minkowski theorem.
In this course we will study quadratic forms (i.e. quadratic polynomials in several variables) and the corresponding bilinear forms. We first classify quadratic forms over the p-adic numbers and over the reals. Assembling all the information over the different Qp’s and over R enables us to study quadratic forms over Q and to prove the Hasse-Minkowski theorem.
Textbook: Short notes and exercices in https://www.mathi.uniheidelberg.de/~khuebner
Suggested references:
1. Jean-Pierre Serre. Cours d’arithmétique. Presses Universitaires de France, Paris, 1977. Deuxième édition revue et corrigée, Le Mathématicien, No. 2
2. J.-P. Serre. A course in arithmetic. Springer-Verlag, New York-Heidelberg, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7
3. T. Y. Lam. The algebraic theory of quadratic forms. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1980. Revised second printing, Mathematics Lecture Note Series
Language: EN