Automorphic forms and periodic functions A lot of resources I looked at, including Wikipedia, thinks of automorphic forms as some sort of generalized periodic functions with respect to the action of discrete groups satisfying some extra conditions (growth conditions, eigenfunction properties, etc.). What comes after as a "prime example" is always something related to modular forms, Poincare half disk, Eisenstein series, etc; the group considered is typically something like $SL(2,\mathbb{R})$.
Is there a reason why the "trivial" example of automorphic forms is not, for instance, simply about eigenfunctions of the Laplacian operator on $\mathbb{R}\times S^1$? Loosely speaking, the cylinder is obtained by discrete action as a coset space $\mathbb{R^2}/\mathbb{Z}$, so the eigenfunctions of the Laplacian operator in two-dimensional Euclidean plane becomes periodic functions in the cylinder.
The only thing I can think of is that either (1) this example is so trivial that it is not worth anyone's time, or (2) the usual "periodic boundary conditions" on the wave equation (equivalent to the action of discrete group $\mathbb{Z}$ on the plane) does not satisfy the requirements of automorphic form framework.
I am new to automorphic forms, so if (1) is true, I would appreciate an explicit description of how the "trivial" example fits into the language of automorphic forms (e.g. how should I think of the generators, growth conditions, etc.), and how (possibly) this fits well into covering space theory. If (2), I would like to know what failed in the construction. Essentially I want to know why, if (2) is true, we should think of automorphic forms as generalized periodic functions at all.
 A: In my opinion, yes, any sort of periodic function is a kind of "automorphic form/function", because they are functions (or sections of vector bundles, or ...) on quotients $\Gamma\backslash G$ or $\Gamma\backslash G/K$ for topological groups $G$ (Lie groups, adele groups, ...) and discrete subgroups $\Gamma$ (and maybe compact subgroups $K$).
The most-classical ideas about periodic functions, say on $\mathbb R$, of period $1$, or of period $2\pi$, or... already have substantial analytical challenges, but the group theory and the associated differential operators, etc., are not the difficulties. Most often, people "doing" automorphic forms are not focusing on analytical difficulties, but the group-theoretic ones, so the very simple classical cases (Fourier series, Fourier transforms) don't shed light.
Also, although I do not claim to be expert on this, I have the impression that the representation-theoretic and harmonic-analytic behavior of unipotent/nilpotent Lie groups (for example) is significantly different from that of reductive groups, such as $SL_2$ or $GL_2$.
Then $SL_2$ is really the smallest semi-simple or reductive group to consider, if we pass over abelian groups and nilpotent/solvable/etc groups.
And the Harish-Chandra isomorphism for the center of the universal enveloping algebra only really applies to semi-simple (or adapted to reductive). Similarly, results about Hecke algebras of various sorts. So there simply aren't good or illuminating analogues in those other cases.
But, yes, I would claim that many basic (and significant) analytical issues are indeed already non-trivial in the classical, abelian cases. Still, their description in most elementary terms will look different from the reductive case.
