# How many proofs of the Weil conjectures are there?

I hope this this is not seen as too much as jumping on the band-wagon, but here goes.

Deligne's proof of the last of the Weil conjectures is well-known and just part of a huge body of work that has lead to prizes, medals etc (wink wink). The other conjectures were proved by Dwork and Grothendieck. According to Wikipedia, Deligne gave a second proof, and then mentions three more proofs. However, it is unclear from what I read as to which of the conjectures they all prove - namely all of them, or just the Riemann hypothesis.

So how many proofs of the complete set of Weil conjectures are there?

• Deligne's l-adic proofs are really only variants of the same proof. Perhaps Kedlaya's p-adic proof is genuinely different. – anon Mar 20 '13 at 23:05
• "... or just the Riemann hypothesis." :) – Martin Brandenburg Mar 20 '13 at 23:50
• What is your equivalence relation on proofs? If someone came up with an ur-proof that specialized to multiple existing proofs, would that increase or decrease your count? – S. Carnahan Mar 21 '13 at 4:18
• @Scott - Ideally an answer would not just state whether proofs are "the same" or not, but be informative as to how they differ or are similar. :-) @Martin - you know what I mean. – David Roberts Mar 21 '13 at 6:53
• I would vote up Scott's comment 5 times if I could. His comment is a proof that the question makes no sense! – user30035 Mar 21 '13 at 7:43

I guess that "just the Riemann hypothesis" means the statement about the eigenvalues of Frobenius acting on the $H^i$ of a smooth proper variety, and "all of then" means the full theory of weights : definition of mixed sheaves, how the 4 operations affect weights etc (from which you get the hard Lefschetz theorem, the decomposition theorem...).

So here is my understanding :

• Deligne's first proof : just the Riemann hypothesis.

• Deligne's second proof : the full package for $\ell$-adic complexes, using the same kind of ideas as his proof but improving them.

• Laumon's proof with the $\ell$-adic Fourier transform : It allows to simplify parts of Deligne's second proof, but does not replace all of it. More precisely, Laumon gives a shorter proof of theorem 3.2.3 of Deligne's "La conjecture de Weil II" that doesn't use section 2 of that paper, but he still needs results from section 1 (the calculation of the monodromy of lisse sheaves on curves in 1.8 and the fact that "most" irreducible lisse sheaves are pure proved in 1.5.1). Once you have theorem 3.2.3, it is pretty easy to deduce the rest of the "full package" from it, so Laumon doesn't say anything about that.

• Katz's proof : Katz gives a simpler proof of theorem 3.2.3 of Deligne's Weil II paper, or rather of a weakening of it that is enough to deduce the Riemann hypothesis and hard Lefschetz. It is not enough to deduce the full package immediately, although this can be done without too much pain. I am not very familiar with Katz's proof. It seems to be different from Deligne's and Laumon's proofs but to use the same kind of techniques as Deligne's proof. My impression is that it is more concrete that Deligne's proof. (Note : I am not convinced that the difference between "just the Riemann hypothesis" and "the full package" is so big. The reason is that, if you know the Riemann hypothesis, then you should be able to deduce the full package for complexes of geometric origin. But global Langlands tells us that, over a smooth curve over a finite field, every irreducible lisse sheaves is of geometric origin up to twisting by a rank 1 sheaf. So in a way everything is of geometric origin. The situation is very different over a number field, of course.)

• Kedlaya's proof : It is supposed to be modeled on Deligne's second proof modulo the simplification of Laumon, but of course the actual technical are different because Kedlaya uses p-adic coefficients. Also, he got the Riemann hypothesis but not the full package, because at the time the theory of relative p-adic coefficients was not fully developed. But now it is !

• Abe and Caro's proof : See http://arxiv.org/abs/1303.0662 They develop the theory of weights for overholonomic D-modules with Frobenius structure on varieties over finite fields and obtain a full Weil II package for them (and even the Asterisque 100 extension, ie the theory of weights for perverse sheaves). They say in the introduction that their method is independent from Kedlaya's method.

• Welcome to MathOverflow! – Akhil Mathew Mar 21 '13 at 23:11
• Perhaps in connection to Kedlaya's proof, one should also mention Faltings' sketch of a crystalline version of Weil II (using convergent F-isocrystals, and logarithmic geometry) in the Grothendieck Festschrift. – anon Mar 22 '13 at 13:56
• Brilliant, this is the sort of thing I was after, especially the approaches not detailed in Wikipedia, and their provenance. – David Roberts Mar 23 '13 at 1:00