Page "Alexander Grothendieck" Paragraph 35

Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's that there is a deep connection between the topological characteristics of a variety and its diophantine ( number theoretic ) properties.

For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.

Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck.