The rank one case is exceptional in that there are lots of nonarithmetic irreducible lattices. No one was saying that it isn't important.

@Jim In order to define the unipotent radical, you need to know enough about unipotent groups to answer the question. And connectedness is not a problem in any characteristic, at least, not if you are talking about algebraic group schemes.

Yes, see for example Milne's notes AGS, XV, 2.5.

This is in Mumford's book on Abelian Varieties (for a finite subgroup scheme). Mumford assumes that the ground field is algebraically closed, but the proof doesn't need this. Alternatively, if you are willing to assume that the ground field is perfect, then the statement follows from the algebraically closed case + descent.

It's a good book, but it takes 327 pages to get to metrics. For someone wanting to learn differential <i>geometry</i>, there are faster routes.

Intersection cohomology (in the context of the etale topology) generalizes the usual l-adic cohomology. With the appropriate choice of the perversity, intersection cohomology gives the usual l-adic cohomology, with or without compact support.

According to the Tate conjecture, l-adic realization gives an equivalence of categories from motives tensor $Q_l$ to the category of l-adic Galois representations generated by the cohomology of smooth projective algebraic varieties over $Q$. The standard conjectures imply the first is semisimple, hence also the second.

I'd guess that, by using etale cohomology, you can give an algebraic proof valid over any algebraically closed field (use the same argument you use over $\mathbb{C}$).

The question should be, why do we need to decompose the cohomology into its primitive parts? You answered your own question: in order to be able to state the Hodge index theorem, or, more generally, Grothendieck's standard conjectures.

My recollection is that they worked with projective varieties, but removing a lower dimensional subvariety is not going to change much. I suggest you look at their article --- it is quite short and readable.

Crystalline cohomology is a Weil cohomology, which explains its importance, but is not very useful for the things I mentioned in my answer. There are relations between flat cohomology and crystalline cohomology, but they are rather complicated to explain. As far as I know, only for the Zariski topology, the etale topology, and topologies in between, do we have a really explicit description of the "points".