Automorphic Forms
and
the Arthur-Selberg Trace Formula

(MA 842)

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 written 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).

References

For most of the background on modular/automorphic forms and Eisenstein series you can take a look at the first few sections of the notes form a class I taught at Columbia.

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
Rankin-Selberg method

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.