Weil's Conjectures and Grothendieck's Standard Conjectures

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Weil's Conjectures and Grothendieck's "Standard Conjectures"


Grothendieck claimed that his "standard conjectures" imply the Weil conjectures. He showed the proof to a class that he taught one summer in the 1960's and he asked one of his students, Kleiman, to write it up and publish it. Kleiman did just that in 1968.

I'm having trouble understanding how the standard conjectures imply the Weil conjectures. Can anybody help me out here?

(Notes for beginners: 1) That's Alexander Grothendieck and Andre Weil. Weil is pronounced like as if it was spelled "vay" and rhymied with "way." 2) If you don't have a clue what I'm talking about and you would like to, here's the wikipedia page. Wikipedia also has excellent pages on "motifs" or "motives" that I mention further down.)


I've been having trouble with this for the last six months or so. All the literature I've been able to find refers to Kleiman's original 1968 paper "for the details" and as far as I can tell, none of the literature I've found so far even gives a hint as to how one might do such a derivation.

I don't normally have acces to a university library, but I did get to one a couple of months ago and looked up Kleiman's 1968 paper on the subject. It is long and dense and as far as I can tell it only spends the last two or three pages actually showing the Weyl conjectures from the standard conjectures. The derivation was completely incomprehensible to me. I thought I had gotten an excellent intuitive feel for cohomology theory during my first three years of graduate school when I was in Michael Kevaire's differential topology group at the Courant Institute in NYU, and that was the tool used, but I could still not extract any intuitive content from the equations. Nothing, nada, zip.

To be fair, I only spent an hour or two with Kleiman's 1968 paper, meaning I really zipped through the first part, and the library was closing when I got to the last few pages, but, still, I've never found any andvanced mathmatics paper that incomprehensible in the last 10 years or so. (I've been doing mathematics avidly for 50 years now.)

I'm on the verge of buying "Dix Exposees de Geometriee Algebrique" which is (more or less) the title of the volume that contains Kleiman's 1968 paper, and I'm considering buying it and working through it, along with the other papers that are alos in there (at least Kleiman's is in English - the others are in French), but I'm hoping that somebody knows of a more recent and better - or simplified - exposition.

Even Kleiman in his paper on "motives and the standard conjectures" in volume I of the conference proceedings "Motives" (1994) gives no hint how to do that derivation and refers to his original paper "for the details."

I've been trying to understand motives (also called motifs) for over a year now, and, Grothendieck invented motives as a tool to lock down the derivation of the Weil conjectures from the standard conjectures. Not understanding how those three things tie together is the biggest single hole in my understanding of motives, and it is more and more of a stumbling block for me all the time.

I recently got a copy of Andre Yves "Une Introduction Aux Motifs." My French is too rusty to really read it yet, but I did search it looking for something on the standard conjectures and the Weyl conjectures. I only found a couple of pages, and pouring over them, it did not look as if Yves gave enough of a hint to really get started on understanding the connection.

Yves book is supposed to be the most elementary and the best introduction to the theory of motives in the literature. At least, there are a lot of number theory professors who say that it is.

The only start that I have is to consider that the factors in the Euler products that appear in the Weyl conjectures are somehow related to the factors that pure motives are known to decompose into if the standard conjectures are true. Did I get that right?

Oh, oh, one more thing. I think the standard conjecture that is ageneralization of the Lefschitz structure for the Lefschitz fixed point theorem gives a series of cycles that might correcpond to the break-down of a pure motive into the parts that correspond to the factors in the Weil conjectures. Did I get that right?

Oh, and I should mention that when I say Weil conjectures, I mean particularly the last conjecture that Deligne proved in 1974 and that corresponds to the classic Riemann hypothesis in analytic number theory.

I would appreciate any help or helpful suggestions.

Thanking you in advance,


P.S. All dates are publication dates, not necessarily corresponding with the year the work was finished.

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