Tags: ? Refs: modular form automorphic representation

Automorphic form

Idea

Cusp forms

Definition: An automorphic form \(f\) on \(G\) is called a cusp form if, for any parabolic \(k\) subgroup \(P=M N\) of \(G\), the \(N\)-constant term \begin{align*} f_{N}(g)=\int_{N(k) \backslash N(\mathbb{A})} f(n g) d n \end{align*} is zero as a function on \(G(\mathbb{A})\).

Relation to \(L^2\) functions and Schwartz space: See also cuspidal representations # Notes

Representations

From https://www.math.arizona.edu/~swc/aws/2022/2022EischenNotes.pdf: