# Fourier expansion of automorphic forms

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.

## 1 Answer

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

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:

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

• Perhaps giving some references (notes by Cogdell etc.) could help the OP (I am in a hurry, so I cannot compile a list now). – GH from MO Oct 21 '15 at 14:55
• @GHfromMO Sure, will do. – Myshkin Oct 21 '15 at 15:06
• One should be careful: in groups other than split $GL_n$, not all automorphic forms/representation "have Whittaker models", e.g., holomorphic Siegel modular forms do not. Sure, there is still a "Fourier expansion" along the Siegel-type maximal proper parabolic, but there are many more Fourier coefficients than Hecke eigenvalues... So I'd say that the "yes" has to be "up to a point..." – paul garrett Oct 21 '15 at 21:16
• Thank you for your remark @paul garrett. I am actually interested in $\lambda_{Sym^rf}$ which is an automorphic form in $GL_{r+1}$ so the answer is certainly yes in this case. – Khadija Mbarki Oct 22 '15 at 7:34