TITLE: Must automorphic forms be square-integrable modulo the center?
QUESTION [8 upvotes]: I've recently needed to learn the basics of automorphic forms and automorphic representations. I've seen two apparently different definitions of automorphic forms, and I'm wondering which is more standard, or if they are in fact equivalent. 
In Bump's book, he defines an automorphic form for $\mathrm{GL}_n/F$, $F$ a number field, as a measurable function $\varphi:\mathrm{GL}_n(F)\setminus \mathrm{GL}_n(\mathbf{A}_F)\rightarrow\mathbf{C}$ which, among lots of other conditions, is square-integrable modulo the center of $\mathrm{GL}_n(\mathbf{A}_F)$, in the sense that
$\int_{Z(\mathbf{A}_F)\mathrm{GL}_n(F)\setminus\mathrm{GL}_n(\mathbf{A}_F)}
\vert \varphi(g)\vert^2dg<\infty$
(the absolute value of $\varphi$ is well-defined modulo the center because $\varphi$ is assumed to have a unitary central character).
On the other hand, in the books of Goldfeld-Hundley, no such condition of square-integrability is imposed in the definition of an automorphic form.
So, which of these (apparently) different conditions are more standard? Or do the other conditions in the definition of an automorphic form imply the square-integrability modulo the center?

REPLY [2 votes]: When I was trying to sort this stuff out and learn all the adjectives, I found the 
Borel--Jacquet article in Corvalis incredibly useful.
Thus: an automorphic form (on $\Gamma\backslash G$ for a congruence subgroup of a real group $G$; the adelic case is a mild modification which Borel--Jacque also explains) is a smooth function of moderate growth which is is $Z(g)$-finite.
It can also be helpful to impose $K$-finiteness.  This is okay because Harish-Chandra proved that automorphic forms (without $K$-finiteness) are an admissible
$G$-rep'n (the different $K$-types appear with finite multiplicity), so that 
passing back and forth between closed $G$-subreps. and $(g,K)$-submodules of the $K$-finite vectors (by taking $K$-finite vectors or taking closures, resp.) is
a bijective process.
One can then speak of $L^p$-automorphic forms, etc., just by imposing the relevant condition (on top of smoothness and $Z(g)$-finiteness).  Cuspforms are automatically $L^2$.
Focusing on $L^2$ is useful for analytic considerations like developing the theory
of Eisenstein series.  It also shows that cuspforms are semisimple as a $G$-rep. (using the Hilbert space structure), while more general automorphic forms need not be.  To see how the $L^2$ theory can be leveraged to apply to the general (i.e. not nec. $L^2$) theory, you can look at Langlands's article following Borel--Jacquet's.  
Franke's artilce on weighted $L^2$ spaces (proving the general form of Eichler--Shimura, for arbitrary groups $G$) also illustrates how to relate $L^2$ contexts
to the context of general automorphic forms.
As Paul Garret says, it is probably good not to be too doctrinaire about all
this, but having at least one set of defintions straight can be very helpful,
and I found Borel--Jacquet invaluable for doing this.  (And I now feel slightly
uneasy whenever someone defines automorphic forms to be $L^2$.)