Notes on global function fields

Question

Does anyone happen to know a good set of notes on global function fields? In particular, I'm hoping to find something akin to Neukirch's chapter on Riemann-Roch theory, but for global function fields rather than number fields. If it helps, I think it relates to understanding this answer - i.e. I need to understand more about the places above $\infty$.

Answer 1

What is it about places above $\infty$ in a finite extension of $K(x)$ compared to places above some other place in $K(x)$ (all of them being trivial on $K$) that you feel you need to understand better? If you are having trouble regarding the place at $\infty$ on $K(x)$ as being on the same footing as the other places on $K(x)$, think about how the linear fractional transformations of $K(x)$ like $x \mapsto 1/x$ or $x \mapsto x/(x+1)$ permute the places on $K(x)$ with residue field degree $1$ over $K$.

Can you think about places (including ramification and residue field extensions) in a more local way, not relying on the crutch of some global ring such as $K[x]$ in $K(x)$? If $E/F$ is a finite extension of fields, a place (discrete valuation) $v$ on $F$ extends to finitely many places of $E$, and in terms of prime ideals this can be studied with the local ring $\mathcal O_v$ inside $F$ and prime ideals in the integral closure of $\mathcal O_v$ inside $E$.

Maybe some references on the MSE pages here or here would be helpful.