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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?
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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)
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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
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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
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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
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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
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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
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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.
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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.
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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. -
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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?
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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
430:-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. --
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subscheme would of course also be finite, so, by the
Nullstellensatz, it would correspond to a finite intersection of maximal ideals in K (here
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better lead that explains the (incredible!) link between enumerative questions in positive characteristics and topology in characteristic zero
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Who proved these? Deligne? Others? Many others? Deligne seems to have gotten a Fields medal for this or at least something related.
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I think that, in the second version of the functional equation, the $ T$ on the left hand side should be replaced by $ T^{-1}$ .
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680:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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I think the conjectures was made mainly by taniyama. you should mention the fact it is now calld shimura-taniyama conjectures.
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is the dimension of the ambient affine space). But such an ideal would be zero-dimensional, which is a contradiction.
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WC4 predicts the degree sequence will be 1,0,2,0,3,0,..,n, but the Betti numbers are 1,0,1,0,,,1?
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Better organisation of the text with statement and explanation of the conjectures before history
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From any perspective you look at it, a curve always has infinitely many non-rational points.
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No, you are mistaken about that. These Weil conjectures were made by Weil around 1949. The
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Last edited at 20:17, 13 June 2007 (UTC). Substituted at 02:40, 5 May 2016 (UTC)
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Much fuller explanation of the conjectures and their cohomological interpretation
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No, what I meant is that it is not I-adic, as you wrote (no italic format).
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Applications and major implications (in number theory, coding theory,...)
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By the way, you should make an account! After you do, come join us at
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finite fields or algebraically closed fields of prime characteristic?
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is an algebraically closed field containing the ground field and
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I guess the traditional typography with â„“ really references the
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Explanation of relations to motives and standard conjectures
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It is? IE, Firefox, and Chrome are all showing it to me as
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http://www.fact-index.com/t/ta/taniyama_shimura_theorem.html
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I moved the page here because it is a group of conjectures.
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https://en.wikipedia.org/Weil_conjectures#Projective_space
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study of variations of Hodge structures on families.
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670:The comment(s) below were originally left at
628:Without Lubkin this article is incomplete. -
222:was made about the time of the conference in
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388:over â„“? Ditto for other articles mentioning
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465:in relation to curves over finite fields
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184:I think that you're right. Thanks!
38:It is of interest to the following
759:High-priority mathematics articles
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678:several discussions in past years
606:Knowledge:Wikiproject Mathematics
599:space has infinitely many points.
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716:Projective space Betti numbers
673:Talk:Weil conjectures/Comments
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333:20:32, 20 November 2017 (UTC)
289:14:24, 9 September 2006 (UTC)
109:and see a list of open tasks.
754:C-Class mathematics articles
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618:02:25, 28 January 2012 (UTC)
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220:Taniyama-Shimura conjecture
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376:-adic or â„“-adic? Capital
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704:Lots of basic copyediting
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728:Where is the error?
121:mathematics articles
409:golfball typewriter
158:Functional equation
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638:—Preceding
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164:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
748:Categories
392:-adic. --
261:Mike Artin
226:, in 1954.
732:Darcourse
206:LittleDan
624:Lubkin !
478:unsigned
321:unsigned
166:unsigned
686:Need:
640:undated
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286:Myasuda
139:on the
30:C-class
513:, let
372:Is it
36:scale.
248:Gauge
224:Nikko
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652:Ozob
634:talk
614:talk
610:Ozob
495:talk
486:talk
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346:Rich
329:talk
237:Who?
200:Move
190:talk
186:Ozob
174:talk
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