I am not a specialist in automorphic forms, can someone explain to me typical elements of adelic Schwartz class, $\mathcal{S}(\mathbb{A})$. Over the real numbers there are obviously elements like:
$$ p(x) \,e^{-x^2} $$
where $p(x)$ is polynomial. My notes have that Schwartz class over adeles is the tensor product over all places.
$$ \mathcal{S}(X\_\mathbb{A}) = \bigotimes\_v \mathcal{S}(X\_v) $$
where $X$ is just the line. So really I would just like to understand better, what are elements of the Schwartz space over $p$-adic numbers?



Adelic Schwartz class
For concreteness, let me just take the adeles of $\mathbb{Q}$.
Let us call a function $f$ on $\mathbb{A}\_{\mathbb{Q}}$ elementary or factorizable if it can be written as
$$
f(x\_{\infty},x\_2,x\_3,x\_5,\ldots)=g\_{\infty}(x\_{\infty})\times \prod\_{p\ {\rm prime}}
g\_{p}(x\_p)
$$
with


1. $g\_{\infty}$ in the usual Schwartz space of $\mathbb{R}$, i.e., smooth with fast decay at infinity and likewise for all its derivatives.
2. for all $p$, $g\_p$ is in the Schwartz-Bruhat space for $\mathbb{Q}\_p$, i.e., locally constant and compactly supported.
3. for all except at most finitely many $p$'s, $g\_p$ is the indicator function of $\mathbb{Z}\_p$.


The space $S(\mathbb{A}\_{\mathbb{Q}})$ of Schwartz-Bruhat functions on the adele ring $\mathbb{A}\_{\mathbb{Q}}$ is the space of all finite linear combinations of elementary functions as above.


Note that number theory references often do not tell what the topology is on this space. It turns out that as a topological vector space $S(\mathbb{A}\_{\mathbb{Q}})$ is isomorphic to the space $\mathcal{D}(\mathbb{R})$ of smooth compactly supported test functions on $\mathbb{R}$.


