A Paley-Wiener theorem and Arthur's trace formula

Modular forms may be recast and generalized as automorphic representations, which are objects of abstract harmonic analysis. The trace formula is a theorem in harmonic analysis which allows one to compare automorphic representations. The Paley-Wiener theorem is also a theorem in harmonic analysis. It characterizes the Fourier transform of smooth compactly-supported functions, and is essential in the proof of Arthur's trace formula. We shall expand on each of these statements, highlighting new results. This is joint work with P. Delorme.Modular forms may be recast and generalized as automorphic representations, which are objects of abstract harmonic analysis. The trace formula is a theorem in harmonic analysis which allows one to compare automorphic representations. The Paley-Wiener theorem is also a theorem in harmonic analysis. It characterizes the Fourier transform of smooth compactly-supported functions, and is essential in the proof of Arthur's trace formula. We shall expand on each of these statements, highlighting new results. This is joint work with P. Delorme.