I am thinking about the computation of the class group and the Picard group for the case of
- Number fields over $\mathbb{Q}$ and $\mathbb{F}_p(t)$
- Complex varieties
I would like to know what kinds of algorithms exist for the computation of these two groups. Evidently Mathematica has some tools in this regard. Is it true that these are not computable in general? In this case, I would like to know about some of the more successful approaches in calculating them.