Question about motives (motifs) and the Weil and Grothendieck conjectures
Deacon John
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 an dpublished it in 1968.
I'm having trouble understanding how the standard conjectures imply
the Weil conjectures. Can anybody help me out here?
(Notes for students: 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.)
http://en.wikipedia.org/wiki/Standar...gebraic_cycles
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 Weil 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 one of the primary tools that we 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 Exposes 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 "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,
John Gayle Aiken IV, PhD 1972, LSU, Baton Rouge
P.S. All dates are publication dates, not necessarily corresponding
with the year the work was finished. All spelling is suspect.
J.S. Milne
> All,
>
> 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 an dpublished it in 1968.
>
> I'm having trouble understanding how the standard conjectures imply
> the Weil conjectures. Can anybody help me out here?
In his orginal article (which you can find at
http://www.math.jussieu.fr/~leila/grothendieckcircle/) Grothendieck
just says (p197) that the standard conjectures imply the Weil
conjectures "by certain arguments of Weil and Serre". I think he meant
(either or both of) the following:
(a) The standard conjectures allow you to define a category of motives
to which exactly the argument Weil used to prove the Weil conjecture
for abelian varieties can be applied (for Weil's argument, see for
example, Section 19 of my 1986 article on abelian varieties).
(b) The standard conjectures allow you to transfer the arguments in
Serre's article Annals of Math 1960 to the context of varieties over
finite fields. This is written out at the end of Kleiman's article.
J.S. Milne
J.S. Milne
that (a) is written out in Proposition 2.48 of my article on Motives
over Finite Fields in the Proceedings of the Seattle conference
(1994). All my papers can be found on my website (google milne).
Deacon John
Professor Milne,
I took your recommendation and read the proofs in the two papers
on your web site that you recommended to me. Your comments and
the proofs in your papers helped me understand the connection
between the Weyl conjectures and the Grothendieck conjectures
and motives.
In case anybody else is interested in following this trail,
I would like to mention that the information in the "Addendum"
file that you have posted on your web site was also very
helpful.
Thank you so much,
Deacon John