In the Wikipedia article it states that Ramanujan's tau conjecture was shown to be a consequence of Riemann's hypothesis for varieties over finite fields by the efforts of Michio Kuga, Mikio Sato, Goro Shimura, Yasutaka Ihara, and Pierre Deligne. Do their papers consist of the only published proof of this result? And is this proof of a similar level of difficulty to Deligne's proof of Riemann's hypothesis?
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1$\begingroup$ The proof of Weil => Ramanujan appears in Deligne's Seminaire Bourbaki 355 (incidentally the paper where he constructs the Galois representations associated to higher weight modular forms). It uses not much more than basic properties of étale cohomology. $\endgroup$– user1594Commented Mar 24, 2010 at 1:49
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$\begingroup$ You may find something useful in Katz, An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields, which appears in Proc Symp Pure Math XXVIII, Mathematical developments arising from Hilbert problems (1974) 275-305, MR0424822. According to the review, the author discusses applications to the Ramanujan-Petersson conjecture. $\endgroup$– Gerry MyersonCommented Mar 24, 2010 at 2:18
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$\begingroup$ Have you tried the nice expository article "Modular forms, the Ramanujan conjecture, and the Jacquet-Langlands correspondence" by J. Rogawski, available on his web page: math.ucla.edu/~jonr/eprints.html ? $\endgroup$– Tommaso CentelegheCommented Mar 24, 2010 at 13:53
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3$\begingroup$ @commenters -- you should have left these as answers! $\endgroup$– Kim MorrisonCommented Mar 27, 2010 at 17:12
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$\begingroup$ Minor nitpick/question over wording: should "consist of" be "constitute"? forum.wordreference.com/showthread.php?t=1501780 $\endgroup$– Yemon ChoiCommented Apr 3, 2010 at 20:12
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1 Answer
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Following the discussion at meta.MO, 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, 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.
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.
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. If you don't worry about this (i.e. you accept that it all works out okay) then the proof is essentially just Eichler--Shimura theory (i.e. the relation between modular forms and cohomology of modular curves), but done with etale cohomology, combined with the Eichler--Shimura congruence relation that connects the $p$th Hecke operator to Frobenius mod $p$. (The latter was treated in the following question.)