$\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$, adeles.

Question

Let $p$ be a prime number. How do I show that $\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$?

Answer 1

The ring in question is the ring of rational functions on the projective line over $\mathbb{F}_p$ which possibly have poles at $0$ and $\infty$. The embedding of $\mathbb{F}_p[T, 1/T]$ in question (the projection of the adelic embedding) is taking corresponding Laurent series expansions at $0$ and $\infty$. To see that this embedding is discrete, one has that a small enough open neighborhood of the zero function in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$ contains only finitely many rational functions that do not have poles outside $0$ and $\infty$, Basically, a rational function in this topology is "small" when it vanishes up to high order at both $0$ and $\infty$, and has no poles. In fact, there is no nonzero rational function that vanishes at just $0$ and has no poles anywhere. Indeed, it is not hard to see that the only rational functions with no poles anywhere on the projective line are the constants.