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?