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, cohomologies, schemes, ...) are due to the desire to access continuity. The benefits of this continuity must be great due to the substantial difficulties in setting it up, but what are they? For example, I have read that Grothendieck's cohomology was crucial in the solution of the Riemann hypothesis for varieties over finite fields, but why is continuity so important for this result?
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closed as not a real question by S. Carnahan♦, Harry Gindi, Loop Space, Anton Geraschenko Jun 1 '10 at 1:34It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 

