the Arthur-Selberg Trace Formula
This course will be focused on the two papers Eisenstein Series and the Selberg Trace Formula I by D. Zagier and Eisenstein series and the Selberg Trace Formula II
by H. Jacquet and D. Zagier. Although the titles of the papers sound
like one is a pre-requisite of the other this actually is not the case.
The main difference between the two is the language (the first is
in classical language whereas the second is written in adelic). We
will spend most of our time on the second paper.
Content of the course: The
course will be self contained (as much as possible). The only
pre-requisite is complex analysis and a little bit of algebraic number theory
(knowing properties of quadratic extensions of the rationals would
suffice, class field theory and Tate's thesis can be helpful,
modular/automorphic forms is a bonus).
We will start with a review of classical (holomorphic) modular forms
and Eisenstein Series on the upper half plane. We will then introduce
Maass forms (for GL(2)), and describe the harmonic analysis of the upper
half plane. I will introduce the Selberg Trace Formula and the Rankin
Selberg method in this setting, which will bring us to the content of
the first paper.
We will then pass to adelic setting. I will start with reviewing
Tate's thesis and Eisenstein series for GL(2) in this setting, which will bring us to the the second paper.
If there is time, we will also dicuss very recent suggestions on
extensions of these ideas to Beyond Endoscopy and related problems.
Very brief summary of the papers:
The basic content of the papers is to derive the Arthur Selberg trace
formula of GL(2) through a Rankin-Selberg integral representation
rather than the classical truncation of Selberg and Arthur. To do so,
both papers express the trace formula as the residue of a function on
the complex plane whose analytic properties are governed by (an
appropriate) Eisenstein series. They then shift contours and `derive`
the trace formula as the residue of the corresponding integral (I will
explain why the the word `derive` is in quotation marks).
Modular forms etc.
- A Course in Arithmetic, J. P. Serre.
Maass forms, Eisenstein series, and the harmonic analysis of the upper half-plane, as well as the Selberg trace formula
- Topics in Classical Automorphic Forms, H. Iwaniec.
Algebraic number theory and Tate's thesis
- Algebraic Number Theory, S. Lang
- Cassels and Frohlich
Time/Location: Tuesday and Thursday 12:30 pm - 1:45 pm, CAS 214
Office Hours: If you have questions etc. please send an email to set a time to meet.