Certain explicit trace formulae on \({\rm GL}_3\)

This is the website for the lecture series given by Masao Tsuzuki from Sophia University (Japan) organized by Héctor del Castillo, Yeansu Kim and Andrei Seymour-Howell, together with the LFANT group.

The lectures will be held at the Storium, Chonnam National University, Gwangju, from April 27 to April 29.

Program

The Arthur-Selberg trace formula is an important tool in the modern theory of automorphic forms. Initially, it was invented by A.Selberg in a general framework of weakly symmetric spaces; By applying an explicit version of the trace formula for the modular forms on the Poincare upper half place, Selberg showed that there are infinitely many linearly independent Maass wave cusp forms on \({\rm SL}_2({\mathbb Z})\). Arthur developed the Selberg's idea in a framework of adelizations of reductive groups over number fields to serve the trace formula as an apparatus to construct the functorial transfers (conjectured by Langlands) of automorphic representations of different groups through comparison of the trace formulae. In this narrative, explicit forms of terms in the trace formulae are not cared about so far because they are not necessary. However, if one would like to apply the trace formula for computational problems in automorphic forms, one need to calculate terms in Arthur-Selberg's trace formula further.

In this series of lectures, we focus on the group \({\rm GL}_3\) and \({\rm GL}_2\), especially for a uniform lattices of \({\rm SL}_3(\mathbb R)\), to explain the idea how an explicit version of the trace formula is deduced from an abstract trace formula. On the way, I would like to give an introduction to the harmonic analysis on a real reductive groups explaining basic concepts like the global characters, the orbital integrals, etc. In the final part of the lecture, I would like to mention an on going project with Werner Hoffmann (Bielefeld) and Satoshi Wakatsuki (Kanazawa) on an explicit trace formula on \({\rm SL}_3({\mathbb Z})\).

• Automorphic forms on \({\rm GL}_n({\mathbb R})\).
• Basic harmonic analysis on \({\rm GL}_n({\mathbb R})\).
• Automorphic forms on \({\rm GL}_n({\mathbb A})\); Arthur's invariant trace formula.
• The weighted characters and the weighted orbital integrals.
• Towards an explicit trace formula for \({\rm SL}_3({\mathbb Z})\).

• Background Lecture I by Andrei Seymour-Howell (LFANT): Introduction to automorphic forms on \({\rm GL}_2\).
• Background Lecture II by Ayan Maiti (LFANT): Automorphic representation of \({\rm GL}_2\).
• Background Lecture III by Héctor del Castillo (LFANT): Abelian trace formula: Poisson summation.

Schedule

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