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?