It appears to me that there are two main ways by which algebraic geometry is applied to number theory. The first is by studying polynomials over fields of number-theoretic interest (which does not seem to be limited to number fields). Diophantine geometry is part of this circle of ideas, as well as the use of elliptic curves to obtain Galois representations, complex multiplication, the Langlands program, and so on.
The second is to study a number ring as a scheme. Some of the work in this area include the work of Voevodsky proving the Milnor conjecture and the Bloch-Kato conjecture, as well as the duality theorems of Artin and Verdier.
My questions are:
- Are the two connected in some way? (Aside from the fact that they are both applications of algebraic geometry of course - perhaps on a related note, I am also thinking of Grothendieck's Dessins d'Enfants and how it would fit into the big picture.)
- What are some modern results on the study of number rings as schemes, aside from the ones mentioned above? What are some good recommendations for literature on this subject? (I have read Algebraic Number Theory by Jurgen Neukirch, which introduces this subject very nicely and develops Riemann-Roch theory for number rings, but I am looking for more, especially more modern work.)
- The etale cohomology of number fields is the same as Galois cohomology, which is the subject of the Milnor conjecture and the Bloch-Kato conjecture. There seems to be more literature on the algebraic geometry (particularly the cohomology) of number fields as compared to number rings (Artin-Verdier duality is one example of a result on the cohomology of number rings). Why is this so?
(I apologize if the title is vague. I find it hard to phrase it in such a way that reflects the nature of the question.)