This is a seminar for advanced Bachelor students and Master students, serving as an introduction to a prominent topic in modern number theory.

Classical modular forms are holomorphic functions defined on the upper half-plane satisfying a plenty of symmetries. Their remarkable properties and spectacular applications identify them as indispensable tools for many areas of mathematics. For instance, modular forms may be used to derive a formula for the number of representations of a natural number as the sum of four (or eight) squares. Most prominently, they also play a crucial role in Andrew Wiles's proof of Fermat's Last Theorem.

The goal of this seminar is to study modular forms from a modern point of view, namely, by means of so-called automorphic representations.

In order to explain what this (roughly) means, we first have to introduce the ring A of (rational) adeles. The idea is that on the rational numbers Q, one has two types of absolute values: On the one hand, there is the "real" (also called "archimedean") absolute value known from Analysis I; on the other hand, any prime number p gives rise to a "p-adic" absolute value on Q (these are called "non-archimedean"). By a theorem of Ostrowski, there are essentially no further absolute values on Q. Now one may form the completions of Q with respect to these absolute values, yielding the fields of real and p-adic numbers. The ring of adeles then results from patching these fields together in a clever way (i.e., in a way preserving their pleasant analytic properties).

Given this, the first step is to regard modular forms as functions on GL_2(A), i.e., on the group of invertible 2x2-matrices with entries in A, having certain invariance properties. Then one may associate to any modular form a representation of the group GL_2(A), and the point is that certain properties of the modular form may be translated into properties of its representation and vice versa. In other words, modular forms may be studied through their associated representations, which enables us to apply the strong machinery of representation theory.

If the number (and interests) of participants permits, we may further touch (maybe one of) the following topics: L-functions associated to modular forms and automorphic representations, the Langlands program, analytic properties of automorphic forms, cohomological aspects of automorphic representations

D. Bump: Automorphic Forms and Representations. Cambridge University Press (1997)

A. Deitmar: Automorphic forms. Springer (2012)

• Organisational meeting/distribution of topics: The preliminary meeting takes place on
  Thursday, February 2nd, at 5-6 pm (sharp) via Zoom. Meeting-ID: 637 9871 3793;
  Password: 235711
• If you would like to participate in the seminar, then please register for the seminar in GRIPS
  (for the link, see "Course homepage"). If you have questions (or if you are unable to
  attend the preliminary meeting), feel free to write me an e-mail: lukas.prader "at"
  ur.de
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk