Talk:Weil conjectures

Functional equation

I think that, in the second version of the functional equation, the $T$ on the left hand side should be replaced by $T^{-1}$. —Preceding unsigned comment added by 84.167.241.29 (talk) 09:44, 14 March 2011 (UTC)[reply]

I think that you're right. Thanks! Ozob (talk) 01:52, 17 March 2011 (UTC)[reply]

Move

I moved the page here because it is a group of conjectures. LittleDan 17:26 12 Jun 2003 (UTC)

I think the conjectures was made mainly by taniyama. you should mention the fact it is now calld shimura-taniyama conjectures. http://www.fact-index.com/t/ta/taniyama_shimura_theorem.html

No, you are mistaken about that. These Weil conjectures were made by Weil around 1949. The Taniyama-Shimura conjecture was made about the time of the conference in Nikko, in 1954.

Charles Matthews 16:29, 20 Sep 2004 (UTC)

Who?

Who proved these? Deligne? Others? Many others? Deligne seems to have gotten a Fields medal for this or at least something related.

Dwork and Deligne are responsible for the proofs themselves, Weil gets credit for the conjectures, and Grothendieck is responsible for creating the theory of etale cohomology which was used by Deligne in his proofs. - Gauge 20:45, 10 July 2005 (UTC)[reply]

Some credit goes also to Mike Artin, I think. Charles Matthews 21:16, 10 July 2005 (UTC)[reply]

Per Allyn Jackson's article [1] on page 1203, it lists B. Dwork's solution of part 1 as being from 1959 (versus 1960 in the article text), and it lists A. Grothendieck as the solver of part 2. It also mentions Grothendieck as having provided a more general solution to part 1 in 1964. Myasuda 14:24, 9 September 2006 (UTC)[reply]

OK, I am probably wrong, as I thought that besides the local Riemann hypothesis all proofs were due to Grothendieck. Actually, I was under the impression (impression I've gotten in reading Katz notes on Deligne's proof and in seeing a conference by Sophie Morel on the subject that can be found in youtube) that both Grothendieck and to some detail Weil had the idea of a rational cohomology of varieties over finite fields that once stated would solve ALL of the conjectures. It turned out that no such thing existed, but - and once again I base my belief mainly in what I've heard experts saying, not so much on my own knowledge - I thought the theory of l-adic cohomology was more than enough to handle all the conjectures on this fashion envisioned by Weil and Grothendieck, except for the local Riemann hypothesis. So, can some one give me an insight? Because, at least the way we see it nowadays, l-adic cohomology makes most of these conjectures easy exercises, and it seems impossible to me that Grothendieck managed to develope the theory without making the proofs. Once again, the impression I got reading on the subject was that he developed the theory to fit the proof, somehow. Or were those previous proofs here mentioned done in some other fashion?

By the way, Deligne was indeed responsible for the proof of one of the conjectures, the local Riemann hypothesis I mention, and it does use the l-adic theory of Grothendieck, but itself introduced a bunch of cool stuff, some of them by Deligne, some by others, such as mixed Hodge structures and the study of variations of Hodge structures on families. — Preceding unsigned comment added by 85.240.65.87 (talk) 20:32, 20 November 2017 (UTC)[reply]

finite fields or algebraically closed fields of prime characteristic?

My knowledge of algebraic geometry is tiny, but it is usu. done over algebraically closed thus infinite fields, isn't it? So I wonder if the current 1st sentence with "...algebraic varieties over finite fields." might be incorrect. Rich 15:43, 23 August 2006 (UTC)[reply]

I've added a para of further explanation. Charles Matthews 08:35, 24 August 2006 (UTC)[reply]

I-adic or ℓ-adic?

Is it I-adic or ℓ-adic? Capital I is indistinguishable from lower-case l, but maybe there's some reason for preferring l over ℓ? Ditto for other articles mentioning l-adic. --Vaughan Pratt (talk) 22:20, 9 April 2011 (UTC)[reply]

I guess the traditional typography with ℓ really references the golfball typewriter. It still helps legibiility to use it, though. It is ℓ-adic everywhere in this article, at least. Charles Matthews (talk) 08:15, 10 April 2011 (UTC)[reply]

It is? IE, Firefox, and Chrome are all showing it to me as l-adic everywhere in this article. I'd fix it myself except that there must be so many articles needing that change that it would be better done by a bot. --Vaughan Pratt (talk) 19:50, 11 April 2011 (UTC)[reply]

No, what I meant is that it is not I-adic, as you wrote (no italic format). Charles Matthews (talk) 21:24, 11 April 2011 (UTC)[reply]

question on rational point in relation to curves over finite fields

Hi, I'm trying to rewrite rational point so that a college freshman can understand some of it. After I wrote the sentence: -"It should be pointed out that many algebraic curves contain no rational points, or just finitely many rational points."-, I put in the seemingly, but not, mickey mouse phrase: -"although they do contain infinitely many nonrational points"- Now I'm not sure that phrase is true for curves over finite fields, unless finite field is used loosely as meaning the completion of a finite field. I would like to keep it or a similar phrase in there to keep it understandable. Is it correct as it stands, or should we put in an exception for finite fields? Thank you.--Rich Peterson 18:04, 27 January 2012 (UTC) — Preceding unsigned comment added by 198.189.194.129 (talk) 198.189.194.129 (talk) 18:29, 27 January 2012 (UTC)[reply]

From any perspective you look at it, a curve always has infinitely many non-rational points.

If you take the viewpoint that a point is a map from the spectrum of a field into a space, then any non-empty scheme has infinitely many points: Take any point P, let k(P) be the residue field at P, and let {k_i} be an infinite family of fields all isomorphic to k(P). These give an infinite family of points. But they're all trivially equivalent; a slightly less trivial example is to let k_i = k(P)(x₁, ..., x_i) where the xs are indeterminates, because then all the points are distinct.

If you take the viewpoint that a point is a geometric point, then you're working over the algebraic closure of the field. Choose an affine open subscheme of the curve and embed it in affine space. If the curve had only finitely many points over an algebraically closed field, then this affine open subscheme would of course also be finite, so, by the Nullstellensatz, it would correspond to a finite intersection of maximal ideals in K[x₁, ..., x_n] (here K is an algebraically closed field containing the ground field and n is the dimension of the ambient affine space). But such an ideal would be zero-dimensional, which is a contradiction.

If you take the viewpoint that a point is a point of the underlying topological space, then you can apply the fact of the previous paragraph: Two non-equivalent geometric points map to distinct points of the topological space, and since there are infinitely many geometric points, the topological space has infinitely many points.

By the way, you should make an account! After you do, come join us at Wikipedia:Wikiproject Mathematics. Ozob (talk) 02:25, 28 January 2012 (UTC)[reply]

Lubkin !

Without Lubkin this article is incomplete. - 2601:4:3880:395:21E:C2FF:FEBB:A1D1 (talk) —Preceding undated comment added 03:18, 4 February 2014 (UTC)[reply]

Could you provide a reference for the facts you're interested in this article having? Or maybe, if you have the time, write a new section of the article yourself? Ozob (talk) 14:54, 4 February 2014 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Weil conjectures/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Need:

better lead that explains the (incredible!) link between enumerative questions in positive characteristics and topology in characteristic zero
Better organisation of the text with statement and explanation of the conjectures before history
Much fuller explanation of the conjectures and their cohomological interpretation
Explanation of relations to motives and standard conjectures
Applications and major implications (in number theory, coding theory,...)
Lots of basic copyediting

Last edited at 20:17, 13 June 2007 (UTC). Substituted at 02:40, 5 May 2016 (UTC)

Projective space Betti numbers

https://en.wikipedia.org/wiki/Weil_conjectures#Projective_space

WC4 predicts the degree sequence will be 1,0,2,0,3,0,..,n, but the Betti numbers are 1,0,1,0,,,1?

Where is the error?

Darcourse (talk) 10:06, 7 May 2022 (UTC)[reply]