In the somewhat old (but extremely nice) reference Adeles and Algebraic Groups by Weil, he cites Riemann-Roch as part of an argument for computing the volume of $\mathbb{A}_k/k$ (paraphrased slightly):
Let $k$ a function field in one variable over $\mathbb{F}_q$, and $\pi$ a uniformizer at a place $v$ of degree $n$. The Riemann-Roch theorem shows at once that the number of cosets of $(\pi)$ in $k_v$ [which he has been using for the local field $\operatorname{Frac}\widehat{\mathcal{O}}_v$] is $q^{g+d-1}$ ($g$ the genus).
Can anyone see what Weil is trying to say here? This argument doesn't make much sense to me for a few reasons.
First, where is he counting cosets in? Taking the simplest possible example of an $\mathbb{F}_q$-point on $\mathbb{P}^1$, it seems like we are asking about the index of $(t)$ in $\mathbb{F}_p((t))$, which is definitely infinite. To get finite-index, it seems like we need to take the index in something integral, say $\mathcal{O}_v$, which just gives the residue cardinality.
Second, the application of Riemann-Roch seems odd. For any divisor $D$ of degree $n$, $q$-exponentiating RR gives $$ [\; \#H^0(D) : \#H^0(K-D) \;] = \frac{\#H^0(D)}{\#H^0(K-D)} = q^{n - g + 1}. $$ None of the obvious choices $D = (\pi), D = - (\pi), K-D = (\pi)$ seem to match his numerics, even putting aside this confusion about where to count cosets in.
