Modul: MAT075 Zurich Graduate Colloquium

## What is... an automorphic form on GL(3,R)?

Talk by Raphael Schumacher

**Date:** 19.10.21 **Time:** 16.30 - 17.30 **Room:**

Around 1980, Jacquet, Piatetski-Shapiro and Shalika published their paper “Automorphic Forms on \(\mathrm{GL}(3)\)” and Bump his book “Automorphic Forms on \(\mathrm{GL}(3,\mathbb{R})\)” in which they founded the theory of automorphic forms on \(\mathrm{GL}(3)\) and proved the functional equation for an \(L\)-function attached to an automorphic representation of \(\mathrm{GL}_3(\mathbb{A})\) for the first time.

As in these two works, we will introduce Hecke-Maass cusp forms \(\phi\in\pi\) on \(\mathrm{GL}(3)\), their first projections \(\phi^1\) and their Fourier-Whittaker expansions. To do this, we have to define
the corresponding Whittaker functions \(W_\phi\), the related dual automorphic forms \(\tilde{φ}\) and their dual Whittaker functions \(\widetilde{W}_\phi\). We will also have to prove some properties of the Fourier-
Whittaker coefficients of Hecke-Maass cusp forms on \(\mathrm{GL}(3)\).

In the end of our talk, we will discuss the various functional equations related to the \(L\)-function on \(\mathrm{GL}(3)\) coming from an arbitrary Hecke-Maass cusp form \(\phi\in\pi\). This is done without the limitation of \(L\)-functions coming only from Maass forms as in Bump’s book.