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Topic: The Birch and Swinnerton Dyer Conjecture

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Birch and Swinnerton-Dyer conjecture - Wikipedia, the free encyclopedia The Birch and Swinnerton-Dyer conjecture is one of the seven Millennium Problems selected by the Clay Mathematics Institute, which is offering a prize of $1 million for a proof of the whole conjecture. In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E, s) at s = 1. It is conjecturally given by a complex formula involving invariants of the curve, studied by Cassels, Tate, Shafarevich and others: these include the order of the torsion group, the order of the Tate-Shafarevich group, and the canonical heights of a basis of rational points. en.wikipedia.org /wiki/Birch_and_Swinnerton-Dyer_conjecture   (991 words)

Poincaré conjecture - Wikipedia, the free encyclopedia As of October 2005, several experts have announced that they have verified Perelman's proof of the Poincaré conjecture, and a consensus seems to be rapidly forming. Whitehead revived interest in the conjecture, when in the 1930s he first claimed a proof, and then retracted it. The Poincaré conjecture is one of the seven Millennium Prize Problems for which the Clay Mathematics Institute is offering a $1,000,000 prize for a correct solution. en.wikipedia.org /wiki/Poincare_conjecture   (1095 words)

PlanetMath: Birch and Swinnerton-Dyer conjecture Coates, A. Wiles, On the Conjecture of Birch and Swinnerton-Dyer, Inv. This is version 5 of Birch and Swinnerton-Dyer conjecture, born on 2003-08-06, modified 2003-08-07. "Birch and Swinnerton-Dyer conjecture" is owned by alozano. planetmath.org /encyclopedia/BirchAndSwinnertonDyerConjecture.html   (222 words)

Research in Algebraic Geometry Determining the rank is a deep open problem which is the object of the conjecture of Birch and Swinnerton Dyer. Much of the work in the field of algebraic cycles is organized around three major conjectures: the Hodge conjecture, the Tate conjecture and the generalized Birch-Swinnerton Dyer conjecture. The Tate conjecture is similar to the Hodge conjecture described above, the difference being that the Tate conjecture is concerned with cohomology classes of algebraic subvarieties of a smooth projective variety M defined over an arbitrary finitely generated field. www.math.duke.edu /~schoen/researchalggeo.html   (712 words)

b-Elliptic-curves Barry Mazur/John Tate/J. Teitelbaum: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Buhler/B. Gross/Don Zagier: On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3. Karl Rubin: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. felix.unife.it /Root/d-Mathematics/d-Number-theory/b-Elliptic-curves   (438 words)

Birch and Swinnerton Dyer Conjecture On the conjecture of Birch and Swinnerton-Dyer for elliptic curves, by Cristian... A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing, Massim... Colwell, Jason (2003-11-18) The Conjecture of Birch and Swinnerton-Dyer for elli... www.scienceoxygen.com /math/761.html   (176 words)

École d'été sur la conjecture de Birch et Swinnerton-Dyer Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog (1966), Séminaire Bourbaki, Vol. The Birch and Swinnerton-Dyer conjecture relates the arithmetic of an elliptic curve to the behaviour of its associated L-series. The Birch and Swinnerton-Dyer conjecture can then be recast in terms of such periods. www.institut.math.jussieu.fr /BSD/programme.html   (1299 words)

Citebase - On Rubin's variant of the p-adic Birch and Swinnerton-Dyer conjecture In:p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 71-80, Contemp. Authors: Agboola, A. We study Rubin's variant of the p-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves concerning certain special values of the Katz two-variable p-adic L-function that lie outside the range of p-adic interpolation. [9] B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0602192   (421 words)

A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing, Massimo Bertolini, Henri Darmon A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing, Massimo Bertolini, Henri Darmon The resulting conjecture can be viewed as an analogue of the classical Birch and Swinnerton-Dyer conjecture, in which $I'(f,K)$ replaces the derivative of the complex $L$-series $L(f,K,s)$ and the circle pairing replaces the Néron-Tate height. A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing projecteuclid.org /getRecord?id=euclid.dmj/1080137206   (502 words)

The Birch and Swinnerton-Dyer Conjecture When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function z(s) near the point s=1. In particular this amazing conjecture asserts that if z(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if z(1) is not equal to 0, then there is only a finite number of such points. Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like www.mi.sanu.ac.yu /~zorano/izazovi21vek/birchsd.html   (164 words)

UCSB Mathematics Graduate Student Seminar Abstract: The now famous conjecture by Birch and Swinnerton-Dyer which was formulated in the early 1960's (in its weak form) states that the order of vanishing at s=1 of the expansion of the L-series of an elliptic function E defined over the rationals, is equal to the rank r of its group of rational points. Largely formulated by Grothendieck and his colleagues, this theory was crucial in proving the Weil conjectures (Deligne) and the Mordell Conjecture (Faltings) on the number theoretic side, and on the geometric side has led to the development of new and powerful methods such as moduli in the theory of curves, surfaces, and higher-dimensional varieties. Soon afterwards, the conjecture was refined to not only give the order of vanishing, but also the leading coefficient of the expansion of the L-series at s=1. www.math.ucsb.edu /~matt/studentseminar_s05.html   (1085 words)

Special Semester on the Birch-Swinnerton-Dyer Conjecture The goal of this seminar is to discuss the notion of Heegner points on modular elliptic curves as well as certain conjectural variants, and the information they provide on the Birch and Swinnerton-Dyer conjecture. In this seminar, I will explain an approach to proving the Main Conjecture of Iwasawa Theory for elliptic curves and for elliptic modular forms in general. The results obtained through this approach, when combined with work of Kato and assuming some expected properties of the Galois representations, prove many instances of the Main Conjecture. www.math.princeton.edu /~ellenber/BSD.html   (593 words)

Qwika - Fonction L One of the most influential examples, and for the history of the most general functions L and for the search for still opened problems, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the first part of the years 1960. It applies to one elliptic curve E, and the problem which it tries to solve is the prediction of the row of an elliptic curve on the whole of the rational numbers: C.has.D. the number of free generators of its group of rational points. As in the case of the very known examples, we can distinguish the representation of series (for example infinite series for the function Zeta de Riemann), and the function in the complex plan which is its analytical prolongation. wikipedia.qwika.com /fr2en/Fonction_L   (577 words)

Fields Institute - Workshop on The Equivariant Tamagawa Number Conjecture Gross' refinement of the Birch and Swinnerton-Dyer Conjecture for CM elliptic curves, and all the conjectures of Chinburg and others in Galois module theory. For abelian number fields this conjecture was proved in 1996 by Nguyen Quang Do, Kolster and Fleckinger up to powers of 2. This Equivariant Tamagawa Number Conjecture (ETNC) encompasses the refinements of various classical conjectures, e.g. www.fields.utoronto.ca /programs/scientific/03-04/tamagawa   (302 words)

Colwell, Jason (2003-11-18) The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order. http://resolver.caltech.edu/CaltechETD:etd-04012004-151307 The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve -- the value of the L-function, to an algebraic invariant of the curve -- the order of the Tate--Shafarevich group. Gross has refined the Birch--Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by the full ring of integers in a quadratic imaginary field. Gross' Conjecture has been reformulated, by Fontaine and Perrin-Riou, in the language of derived categories and determinants of perfect complexes. etd.caltech.edu /etd/available/etd-04012004-151307   (217 words)

Omitted topics Tate showed, for elliptic curves, that the first statement implies the second up to a power of p, which was removed by Milne, and that the conjecture was equivalent to the finiteness of the Tate-Shafarevich group. Two natural approaches to restore the conjecture, which work for curves, are either to insist on non - isotriviality of the variety or to look at points which are not in the image of the Frobenius map. Lang conjectured that, on a variety of general type over a number field, the set of rational points is not Zariski dense. www.ma.utexas.edu /users/voloch/surveylatex/node5.html   (526 words)

École d'été sur la conjecture de Birch et Swinnerton-Dyer This school is devoted to presenting progress in the direction of the Birch-Swinnerton-Dyer conjecture, with an emphasis on results obtained in recent years. It will lead the audience from the statement of the conjecture to the most sophisticated techniques and provide the opportunity to meet some of the best specialists in the field. Elle amènera les auditeurs de l'énoncé de la conjecture à l'étude des techniques les plus sophistiquées et leur donnera l'occasion de rencontrer des spécialistes de premier plan. www.math.jussieu.fr /bsd   (175 words)

large.sha Sure, if you take for granted the formula for S and this part of the conjecture of Birch and Swinnerton-Dyer then it is "fairly obvious" that the parity of Sha B & McG did an exhaustive search for curve with prime conductor, and used the conjecture of Birch and Swinnerton-Dyer to surmise Sha; their curve with Sha=289 had rank zero (as do all the curve with large Sha Examples where this has been proved rigorously would be of greatest interest, but I'd very much like to see cases where this is predicted by the Birch- Swinnerton-Dyer conjecture or the Gross-Zagier formula as well. www.math.niu.edu /~rusin/known-math/95/large.sha   (2129 words)

Tan: Refined theorems of the Birch and Swinnerton-Dyer type TATE, On the conjecture of Birch and Swinnerton-Dyer and a geometric analogue, Séminaire Bourbaki n° 306 ( TEITELBAUM, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. [Tn1] K.-S. Refined conjectures of the Birch and Swinnerton-Dyer Type, Harvard University, Dept. of Mathematics, Ph. www.numdam.org /numdam-bin/item?id=AIF_1995__45_2_317_0   (407 words)

Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer - Rubin (ResearchIndex) 8 the conjecture of Birch and Swinnerton-Dyer (context) - Coates, Wiles - 1977 Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Invent. @misc{ rubin81elliptic, author = "K. Rubin", title = "Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer", text = "K. Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Invent. citeseer.ist.psu.edu /288842.html   (481 words)

Mathematics of Computation Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves J. Buhler, B.H. Gross and D.B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. www.ams.org /mcom/2001-70-236/S0025-5718-01-01320-5/home.html   (1044 words)

About "The Birch and Swinnerton-Dyer Conjecture" The Math Forum is a research and educational enterprise of the Drexel School of Education. The conjecture addresses the problem of enumerating rational points, such as on elliptic curves. A Clay Mathematics Institute Prize problem, with a description in pdf format by Andrew Wiles. mathforum.org /library/view/17101.html   (44 words)

INI Programme RMA Workshop - Random Matrix Theory and the Birch/Swinnerton-Dyer Conjecture This Spitalfields Day concerns the Birch and Swinnerton-Dyer conjecture, which describes a deep connection between the rank of an elliptic curve and the order of vanishing of an L-function. INI Programme RMA Workshop - Random Matrix Theory and the Birch/Swinnerton-Dyer Conjecture Particular attention will be paid to recent work which uses random matrix theory to make precise predictions for the ranks of families of elliptic curves. www.newton.cam.ac.uk /programmes/RMA/rmaw01_spital.html   (318 words)

Amazon.ca: P-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture: A Workshop on P-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture August 12: Books P-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture: A Workshop on P-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture August 12 Amazon.ca: P-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture: A Workshop on P-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture August 12: Books Top of Page : P-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture: A Workshop on P-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture August 12 www.amazon.ca /exec/obidos/ASIN/0821851802   (256 words)

Open Questions: Elliptic Curves and Modular Forms This conjecture is now known to be true for all elliptic curves, but even without that, the Birch-Swinnerton-Dyer conjecture could subsume it to the extent of including the stipulation that L(E,s) is analytic at s=1. This conjecture postulates exactly what one would hope to be true, namely that the L-functions of elliptic curves (and their twists) have the same very symmetric properties as the classical Dirichlet L-functions, namely an analytic continuation and a functional equation. Until recently, the Hasse-Weil conjecture was known to be true in only two cases: when the elliptic curve had the property known as "complex multiplication", and when the elliptic curve had the property of being "modular". www.openquestions.com /oq-ma017.htm   (18524 words)

Henri Darmon - Recent progress in the theory of elliptic curves After Wiles' proof of the Shimura-Taniyama conjecture, the Birch and Swinnerton-Dyer conjecture has become the outstanding open problem in the arithmetic theory of elliptic curves. Recently Bertolini and I have been able to prove part of a p-adic variant of the Birch and Swinnerton Dyer conjecture which applies to curves of higher rank. In the late 80's, the work of Kolyvagin led to an almost complete proof of the Birch and Swinnerton-Dyer conjecture for elliptic curves of (analytic) rank at most one. www.cms.math.ca /CMS/Events/winter98/w98-abs/node2.html   (156 words)

Math JS Milne Preprints Prove the full conjecture of Birch and Swinnerton-Dyer in the case of a constant abelian variety over a global field of prime characteristic; in particular, give the first examples of nonzero abelian varieties whose Tate-Shafarevich groups are known to be finite. Examines the conjecture of Langlands and Rapoport and its consequences; introduces the notion of a canonical integral model. Assuming the Hodge conjecture for abelian varieties of CM-type, one obtains a good category of abelian motives over the algebraic closure of a finite field and a reduction functor to it from the category of CM-motives. www.jmilne.org /math/Preprints   (2414 words)

Citebase - From the Birch & Swinnerton-Dyer Conjecture over the Equivariant Tamagawa Number Conjecture to non-commutative Iwasawa theory - a survey Fukaya and K. Kato, A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, preprint (2003). Huber and G. Kings, Equivariant Bloch-Kato conjecture and non-abelian Iwasawa main conjecture, Proceedings of the International Congress of Mathematicians, Vol. Kleiman, Algebraic cycles and the Weil conjectures, Dix esposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0507275   (779 words)

Tygo Search - Dyer BIRCH AND SWINNERTON-DYER CONJECTURE Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic e École d'été sur la conjecture de Birch et Swinnerton-Dyer PARIS, 4 - 12 juillet 2002 Organisateurs / Organizers D. MEREL Comit École d'été sur la conjecture de Birch et Swinnerton… www.tygo.com /search?s=Dyer&pg=4   (285 words)

Intercity2001 The topic for the Spring 2001 Intercity seminar is the Birch Swinnerton-Dyer conjecture. 15:30-16:30 Jasper Scholten: An introduction to the Birch-Swinnerton-Dyer conjecture. The Birch-Swinnerton-Dyer Conjecture/ Het Vermoeden van Birch en Swinnerton-Dyer www.science.uva.nl /~geer/ic2001.html   (269 words)

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