## What would a “moral” proof of the Weil Conjectures require?

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At the very end of this 2006 interview (rm), Kontsevich says

"...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better proofs...I think the Index Theorem by Atiyah and Singer...its original proof, I think it's ugly in a sense and up to now, we don't have "the right proof." Or Deligne's proof of the Weil conjectures, it's a morally wrong proof. There are three proofs now, but still not the right one."

I'm trying to understand what Kontsevich means by a proof not being "morally right." I've read this article by Eugenia Cheng on morality in the context of mathematics, but I'm not completely clear on what it means with respect to an explicit example. The general idea seems to be that a "moral proof" would be one that is well-motivated by the theory and in which each step is justified by a guiding principle, as opposed to an "immoral" one that is mathematically correct but relatively ad hoc.

To narrow the scope of this question and (hopefully) make it easier to understand for myself, I would like to focus on the second part of the comment. Why would Kontsevich says that Deligne's proof is not "morally right"? More importantly, what would a "moral proof" of the Weil Conjectures entail?

Would a morally proof have to use motivic ideas, like Grothendieck hoped for in his attempts at proving the Weil Conjectures? Have there been any attempts at "moralizing" Deligne's proof? How do do the other proofs of the Weil Conjectures measure up with respect to mathematical morality?

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## 4 Answers

I would guess that Grothendieck's envisaged proof, via the standard conjectures, would be "morally right" in Kontsevich's sense. (Although there is the question of how the standard conjectures would be proved; since they remain conjectures, this question is open for now!)

The objection to Deligne's proof is that it relies on various techniques (passing to symmetric powers and Rankin--Selberg inspired ideas, analytic arguments related to the positivity of the coefficients of the zeta-function, and other such things) that don't seem to be naturally related to the question at hand. I believe that Grothendieck had a similar objection to Deligne's argment.

As a number-theorist, I think Deligne's proof is fantastic. One of the appeals (at least to me) of number theory is that none of the proofs are "morally right" in Kontsevich's sense. Obviously, this is a very personal feeling.

(Of course, a proof of the standard conjectures --- any proof, to my mind --- would also be fantastic!)

[Edit, for clarification; this is purely an aside, though:] Some arguments in number theory, for example the primitive root theorem discussed in the comments, are pure algebra when viewed appropriately, and here there are very natural and direct arguments. (For example, in the case of primitive roots, there is basic field theory combined with Hensel's lemma/Newton approximation; this style of argument extends, in some form, to the very general setting of complete local rings.) When I wrote that none of the proof in number theory are "morally right", I had in mind largely the proofs in modern algebraic number theory, such as the modularity of elliptic curves, Serre's conjecture, Sato--Tate, and so on. The proofs use (almost) everything under the sun, and follow no dogma. Tate wrote of abelian class field theory that "it is true because it could not be otherwise" (if I remember the quote correctly), which I took to mean (given the context) that the proofs in the end are unenlightening as to the real reason it is true; they are simply logically correct proofs. This seems to be even more the case with the proofs of results in non-abelian class field theory such as those mentioned above. Despite this, I personally find the arguments wonderful; it is one of the appeals of the subject for me.

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In that theorem, "the only cyclic groups" should be "the only units groups mod m" and p should be an odd prime. There are two parts to this theorem: (1) units mod m are not cyclic for other m and (2) units mod m are cyclic for those m. It's easier to show (1). If m is any other number then the units mod m have more than one element of order 2: if m is divisible by 4 and is not 4 itself, then -1 and 1+m/2 mod m both have order 2 and are different. If m is twice an odd number, units mod m and mod m/2 are isomorphic, so we can focus on odd m. – KConrad Feb 10 2010 at 3:57

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What problem do you have with the standard proofs? – Qiaochu Yuan Feb 10 2010 at 3:57

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If m is not an odd prime power then we can write m = ab where a > 2 and b > 2, so (Z/m)* = (Z/a)* x (Z/b)*, so visibly there are a few elements of order 2: (-1,1), (1,-1), and (-1,-1). That completes a morally right proof of item (1). – KConrad Feb 10 2010 at 3:58

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As for item (2), that when m is 2, 4, p^k, or 2p^k for odd prime p, the units mod m are cyclic, we can reduce easily to the case of p^k. In this case I'd use p-adics: (Z/p^k)* = (Z_p/p^kZ_p)* = Z_p*/(1 + p^kZ_p) = (mu_{p-1} x (1+pZ_p))/(1+p^kZ_p) = mu_{p-1} x (1+pZ_p)/(1+p^kZ_p). Since mu_{p-1} is cyclic of order p - 1 and 1+p is a generator of the second factor, a group of p-power order, we've written (Z/p^k)* as a product of cyclic groups of relatively prime order, so the group itself is cyclic. QED – KConrad Feb 10 2010 at 4:01

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The only if part is dealt with by Keith Conrad's comment. For the if part: the existence of a primitive root mod p is algebra (any finite multiplicative subgroup of a field is cyclic). To get from mod $p$ to mod $p^n$, one uses Hensel's lemma (or, if you prefer, Newton approximation); of course, you can also be very explicit and just observe directly via the binomial theorem that 1 + p, or 1 + 4 when p = 2, generate the units mod p^n that are 1 mod p (or 1 mod 4 when p = 2). Hensel/Newton is the basic tool for deforming over nilpotent ideals, and I think is "morally right" in that context. – Emerton Feb 10 2010 at 4:04
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I am by no means the expert on algebraic geometry. But maybe I can say a little bit. Kontsevich seems ever wrote a book"Beyond Number" There is one paragraph:

“Very often a mathematician considers his colleague from a different domain with disdain -- what kind of a perverse joy can this guy find in his unmotivated and plainly boring subject? I have tried to learn the hidden beauty in various things, but still for many areas the source of interest is for me a complete mysetery.

My theory is that too often people project their human weakness/properties onto their mathematical activity.

There are obvious examples on the surface: for instance, the idea of a classification of some objects is an incarnation of collector instincts, the search for maximal values is another from of greed, computability/decidability comes from the desire of a total control.

Fascination with iterations is similar to the hypnotism of rhythmic music. Of course, the classification of some kinds of objects could be very useful in the analysis of more complicated structures, or it could just be memorized in simple cases.

The knowledge of the exact maximum or an upper bound of some quantity depending on parameters gives an idea about the range of its possible values. A theoretical computability can be in fact practical for computer experiments. Still, for me the motivation is mostly the desire to understand the hidden machinery in a striking concrete example, around which one can build formalisms.

..... In a deep sense we are all geometers."

I think what Kontsevich mean is that not only the result should be correct, but also the method to get the result should be elegant and natural. Just as he mentioned, the most interesting things to him is the hidden machinery behind the striking examples. For example, say Atiyah-Singer index theorem. Rosenberg ever mentioned in the class, this theorem should have the machinery living in the abelian categories or even exact categories instead of triangulated categories(where Grothendieck Riemman Roch is now living). I guess what they are thinking about is that one should use some universal constructions ,universal theory"(in some sense). They always make emphasis on one sentence "Mathematics should be simple" which might means that the proof of some big theorem should be simple. That is to say, one does not pay much "brain thinking" because "brain of human are weak"

However, I agree with Emerton that this is very personal feelings

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Can you give a reference for the quote? – Jonah Sinick Feb 11 2010 at 5:10

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Found it ("The Unravelers: Mathematical Snapshots" edited by Jean-Francois Dars and Annick Lesne) – Jonah Sinick Feb 11 2010 at 19:42

Presumably, Kontsevich is referencing the fact that Deligne used a "trick" to prove the Weil conjectures. Kontsevich is presumably talking about the Grothendieck standard conjectures on algebraic cycles, which would allow us to "realize the dream of motives".

I see no way to "moralize" Deligne's proof, because as I said, it relies on a "trick" which circumvents the hard parts of the standard conjectures.

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=P Can't vote me down anymore. Oh well. It's quite funny that my answer was essentially the same as Emerton's, and it was posted first. I included fewer details, but that doesn't mean my answer deserves a vote down. – Harry Gindi Feb 10 2010 at 5:39

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Dear Harry: This koan, from the Jargon File, is one of my favorites. "A novice was trying to fix a broken Lisp machine by turning the power off and on. Knight, seeing what the student was doing, spoke sternly: "You cannot fix a machine by just power-cycling it with no understanding of what is going wrong." Knight turned the machine off and on. The machine worked." – Tom Church Feb 10 2010 at 9:11

This is both an answer and a question:

As part of a response to a previous question of mine, David Speyer wrote that:

... it is known how to adapt Weil's proof of the Riemann hypothesis to higher dimensional S, if one had an analogue of the Hodge index theorem for $S \times S$ in characteristic p. I've been told that a good reference for this is Kleiman's Algebraic Cycles and the Weil Conjectures...

So perhaps a "moral" proof would require a Hodge index theorem in characteristic p.

However, David later writes that Grothendieck's standard conjectures assert that the Hodge theorem holds. So is this possible proof the same as "Grothendieck's envisaged" one?

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Yes, I think so. Grothendieck's standard conjectures incorporate certain "positivity" results about cohomology and cup product which suffice to imply RH. You could look at Serre's letter to Weil, in which he proves a version of RH for endomorphisms of a complex projective variety via Hodge theory, to get a sense of how such things might work. And then Kleiman's article gives the details, I believe. – Emerton Feb 10 2010 at 20:28
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