Seminar quadratic forms

Which integers are the sum of two, three or four squares, i.e. which n ∈ ℤ are of the form n = x² + y², n = x² + y² + z² or n = x² + y² + z² + t² with x,y,z,t ∈ ℤ? Answers to these questions were given by Euler, Lagrange and Legendre already in the 18th century.

A quadratic form over the integers is a homogeneous quadratic plolynomial of the form

A natural question is for which n ∈ ℤ is Q(x₁,…,x_d) = n solvable with (x₁,…x_d) ∈ ℤ^d?

This is a relatively hard problem. But asking for rational solutions only, the theorem of  Hasse and Minkowski gives a satisfying answer: There is a rational solution if and onlz if there is a real solution and for each prime number p a p-adic solution.

Quadratic forms and lattices appear in topology, algebraic geometry and have connections to  complex analysis codes, groups and computeralgebra. In 2022 Maryna Viazovska was awarded a Fields medal for her proof that, the E₈ lattice is the densest spherepacking in 8 dimensions. This illustrated that quadratic forms are still an area of active research.

Registration:
The briefing is on March 22. at 10 am in SR 6.

Requirements:

• give a 60 minute talk
• for a seminar certificate provide a written report of 8-10 pages

Prerequisites:
Linear Algebra I

Literature:

1. J.W.S. Cassels: Rational Quadratic Forms, Academic press.
2. J.P. Serre: A course in arithmetic, Springer.
3. M. Kneser: Quadratische Formen, Springer.

See here for links to the literature .