Let $\theta(z) = \sum_{n \in \mathbb{Z}} q^{n^2}$ with $q = e^{2\pi i n z}$. Can we extend the theta function to $p$-adic arguments? Here's an example:
$$ \theta( 1 + p^k) = \sum_{n \in \mathbb{Z}^2} e^{2\pi i n^2 (1 + p^k)} = \theta(1) $$
In my textbook they define a rather unusual upper half plane $\mathbb{H} \simeq SL(2, \mathbb{Z}) \backslash SL(2, \mathbb{R})/SO(2, \mathbb{R}) $ but there is also the adelic way to define the same object: $$\mathbb{H} \simeq SL(2, \mathbb{Q}) \backslash SL(2, \mathbb{A})/SO(2, \mathbb{A}) $$ There is even a more basic question of what elements of $SO(2, \mathbb{A})$ even look like. Perhaps these are rotations of the circle $\{x^2 + y^2 = 1\} \subseteq \mathbb{A}^2$.