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I assume it is partially because they are good generalizations of polynomial rings, but what makes this generalization better than graded algebras or other generalizations of polynomial rings?

For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in understandin …

I'm interested in the arithmetic/diophantine equation applications of arithmetic/algebraic geometry. From what I understand, many of the difficult/technical aspects of the latter theories (sheaves, co …

In the Wikipedia article it states that Ramanujan's tau conjecture was shown to be a consequence of Riemann's hypothesis for varieties over finite fields by the efforts of Michio Kuga, Mikio Sato, Gor …

Using p-adic analysis, Dwork was the first to prove the rationality of the zeta function of a variety over a finite field. From what I have seen, in algebraic geometry, this method is not used much an …

Global fields consist of finite extensions of $\mathbb{Q}$ (algebraic number fields) and finite extensions of $\mathbb{F}_q(x)$ (function fields in 1 variable over a finite field). The latter are isom …