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 understanding arithmetic questions?



Why are topological ideas so important in arithmetic?
Why are topological ideas so important in arithmetic? In some sense KConrad is of course spot on, but let me offer a completely different kind of answer.


Why are complex functions of one variable so important in arithmetic? (Zeta function, L-functions, Riemann hypothesis, Birch--Swinnerton-Dyer, modular forms, theta series, Eisenstein series...).


Why is geometry so important in arithmetic? (Faltings' theorem, applications of algebraic geometry, low-dimensional arithmetic of varieties (elliptic curves etc))


Why is K-theory so important in arithmetic? (Bloch-Kato, Voevodsky...)


Why is logic so important in arithmetic? (Julia Robinson, Matiyasevich, Ax-Kochen and then Hrusovski proving that "if it's true in char p for suff large p then it's true in char 0" in the context of some very deep statements)


Why is functional analysis so important in arithmetic? (L^2 functions on $\Gamma\backslash G$ with $G$ a semisimple Lie group being related to automorphic forms and hence to number theory via Langlands, with crucial analytic tools like the trace formula).


Why are dynamical systems so important in arithmetic? (3x+1 problem, work of Deninger, or of Lind/Ward and their school).


Here's the answer: it's because arithmetic is a very mature subject---it has been around literally thousands of years, and because it has been around so long, there is far more of a chance that someone will come along with an insight relating [insert arbitrary area of pure mathematics here] with arithmetic. So in some sense it's a historical fluke. If we were all born with continuum-many fingers which we could move only in real-analytic ways, and we didn't discover the positive integers until much later on, then arithmetic would be all new and we'd be waiting for Gauss, and real analysis would be as old as the hills, and people would be asking "why is [insert arbitrary thing] so important in real analysis"?


[PS (1) yeah I know, I was being facetious at the end, and (2) yeah I know, my list at the top is woefully incomplete]


