Following the discussion at [meta.MO][1], I'm going to post a good answer from the comments (made by JT) as a "community wiki" answer.  I should mention that the Rogawski article mentioned by Tommaso says almost nothing about the proof of Ramanujan's conjecture, but it seems to be a very nice introduction to Jacquet-Langlands.

Deligne reduced Ramanujan's conjecture about the growth of tau to the Weil conjectures (in particular, the Riemann hypothesis) applied to a Kuga-Sato variety, in his paper [Formes modulaires et representations l-adiques][2], Seminaire Bourbaki 355.  I believe Jay Pottharst has made an English translation available.

Deligne then proved the Weil conjectures in his paper [La conjecture de Weil. I][3].

As far as I know, all known proofs of this conjecture involve the use of cohomology of varieties over finite fields in an essential way.

  [1]: http://tea.mathoverflow.net/discussion/328/should-we-do-anything-if-a-question-is-answered-well-in-the-comments/
  [2]: http://www.google.com/search?q=formes+modulaires+et+representations+l-adiques
  [3]: http://www.numdam.org/item?id=PMIHES_1974__43__273_0

Added by Emerton: One point to make is that the Weil conjectures (in their basic form,
saying that the eigenvalues of Frobenius on the $i$th etale cohomology of a variety over
$\mathbb F_q$ have absolute value $q^{i/2}$) apply only to smooth proper varieties.  On the other hand, the Kuga-Sato variety is the symmeteric power of the universal elliptic curve over a modular curve, which is not projective.  Thus one has to pass to a smooth compactification in order to apply the Weil conjectures, and then hope that this does not mess anything up in the rest of the argument.  A certain amount of Deligne's effort in
his Bourbaki seminar is devoted to dealing with this issue.