Let $X$ be a smooth projective curve over a field $k$. We let $\omega$ be the canonical line bundle of $X$ and we denote by $F$ the field of $k$-valued rational functions on $X$.

(1) When $k$ is algebraically closed then $\omega$ is a dualizing sheaf for $X$. From there it is easy to prove Riemann-Roch for regular (holomorphic) line bundles $L$ over $X$: By this I mean a precise formula which computes the Euler characteristic of $L$ in terms of the degree of $L$ and the genus of $X$ (I think of both as being topological invariants).

(2) When $k$ is a finite field then one may consider the topological ring $\mathbf{A}_F$, the ring of Adeles of $F$. Doing Fourrier analysis on this self-dual locally compact abelian group and doing a counting argument one may deduce Riemann-Roch.

Q1: Is it possible to generalize Riemann-Roch to other fields? What about real and $p$-adic numbers?

Q2: Is $\omega$ a dualizing sheaf when $k$ is finite? If not

(I guess that in general one has to replace the notion of dualizing sheaf by some kind of complex in a derived category)

Q3: Is there a way to prove simultaneously $(1)$ and $(2)$?

Q4: Is there some notion that would encompass simultaneuously $\mathbf{A}_F$ and $\omega$?