Fourier expansion of automorphic forms

Question

we know that for $r \in \{1,2,3,4\},$ $\lambda_{Sym^rf}$ is an automorphic form (here $f$ is a modular form for the full modular group) and this fact is conjectured for $r\geq 5$ by Langlands and Serre. My question is the following: have automorphic forms Fourier expansions? Thanks in advance.

Answer 1

I don't see the relation between the first sentence and the question, so perhaps I'm missing something, but the answer to the question is yes, regardless.

Fourier expansion is a very important tool in the study of automorphic forms, and works for any automorphic form over any reductive algebraic group.

As usual, the cuspidal and non-cuspidal cases are very different, and the $G=\mathrm{GL}_n(\mathbb{A})$ case is particularly well understood.

For instance, for $\mathrm{GL}_n(\mathbb{A})$ and $\varphi$ cuspidal, we have a beautiful theorem of Shalika and Piatetski-Shapiro:

$\varphi$ is globally generic, that is, has non-vanishing Whittaker-Fourier coefficients, and the Fourier expansion is absolutely convergent and uniformly converges on any compact set.

For a nice exposition of the analytic theory on $\mathrm{GL}_n$, see

• J. W. Cogdell, Notes on L-functions for $\mathrm{GL}_n$

As a textbook reference, section 3 of Bump's classic book covers this aspect of automorphic forms:

• Daniel Bump, Automorphic Forms and Representations

A more concrete and advanced reference that I've found very useful:

• Baiying Liu, Fourier Coefficients of Automorphic Forms and Arthur Classification

The original references for the Shalika and Piatetski-Shapiro theorem:

• J. Shalika, The multiplicity one theorem for $\mathrm{GL}_n$ (1974)
• I. Piatetski-Shapiro, Multiplicity one theorems (1979)