250087 VO Advanced topics in global number theory (2024S)

This course is an introduction into the theory of unitary representations of GL(n) over the ring of adeles and their eminent role in modern number theory.

We will start with basic concepts (e.g., What is the ring of adeles?), explain fundamental notions of representation theory of locally compact groups and we will then focus on the case of our interest, the group GL(n) over the adeles.

A prominent role will take the cuspidal spectrum of GL(n) - a unitary representation on a certain space of square-integrable functions, which is of highest number-theoretical importance. We will see - by analogy to GL(1) - that the cuspidal spectrum of GL(n) decomposes over irreducible unitary representations: The cuspidal “automorphic” representations of GL(n).

However, these global automorphic „building blocks“ decompose even further: Namely, they are the infinite tensor product of local irreducible unitary representations, one for each place of the field of rational numbers! This final observation, which will be one of the main objectives of the lecture, will close the loop back to modern number theory.