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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 …

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?

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 …

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 …

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 …