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?
closed as not a real question by S. Carnahan♦, Harry Gindi, Andrew Stacey, Anton Geraschenko♦♦ Jun 1 2010 at 1:34