Birch & Swinnerton-Dyer: Extensive Numeric Evidence: curvesrank>1 (open)
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From: master1729 on 10 Sep 2009 03:29 dear musatov , Birch & Swinnerton-Dyer is the hardest possible math. So , i very strongly advise you not to talk about it.
From: marty.musatov on 10 Sep 2009 04:05 > dear musatov , Birch & Swinnerton-Dyer is the hardest > possible math. > > So , i very strongly advise you not to talk about it. You did not even read the text. There are no 'hardest' math problems. Not eve P=NP is hard. P=NP is easy. Musatov
From: marty.musatov on 10 Sep 2009 04:06
> dear musatov , Birch & Swinnerton-Dyer is the hardest > possible math. > > So , i very strongly advise you not to talk about it. Or what, you might have to admit my accessing algorithm to work through wikipedia from my server through Google has solved it? That is the idea, Sir. {{Millennium Problems}} In [[mathematics]], the '''Birch and Swinnerton-Dyer conjecture''' relates the [[Rank of an abelian group|rank]] of the [[abelian group]] of points over a [[number field]] of an [[elliptic curve]] ''E'' to the order of the zero of the associated [[Hasse-Weil L-function|L-function]] ''L''(''E'', ''s'') at ''s'' = 1. Specifically, it is conjectured that the [[Taylor expansion]] of ''L''(''E'', ''s'') at ''s'' = 1 is :<math>L(E,s) = c(s-1)^r + \text{higher order terms} \, ,</math> where ''c'' is not zero and ''r'' is the rank of ''E'' over the field of rational numbers.<ref name=Wiles>[http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/birchswin.pdf The Clay Math Institute Official Problem Description (pdf)] by [[Andrew Wiles]]</ref> As of 2007, it has been proved only in special cases, all of rank less than or equal to 1. It has been an open problem for around 40 years, and has stimulated much research; its status as one of the most challenging mathematical questions has become widely recognized. It is one of the [[Millennium Prize Problems]] listed by the [[Clay Mathematics Institute]], which has offered a USD 1,000,000 prize for the first correct proof. == Background == In 1922 [[Louis Mordell]] proved [[Mordell's theorem]]: the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite sub-set of the rational points on the curve, from which all further rational points may be generated. If the number of rational points on a curve is [[infinity|infinite]] then some point in a finite basis must have infinite order. The number of ''independent'' basis points with infinite order is called the [[rank of an abelian group|rank]] of the curve, and is an important [[invariant]] property of an elliptic curve. If the rank of an elliptic curve is 0 then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points. Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) these cannot be generalised to handle all curves. An ''L''-function '''''L''(''E'', ''s'')''' can be defined for an elliptic curve ''E'' by constructing an [[Euler product]] from the number of points on the curve modulo each [[prime number|prime]] ''p''. This ''L''-function is analogous to the [[Riemann zeta function]] and the [[Dirichlet L-series]] that is defined for a binary [[quadratic form]]. It is a special case of a [[Hasse-Weil L-function]]. The natural definition of ''L''(''E'', ''s'') only converges for values of ''s'' in the complex plane with Re(''s'') > 3/2. [[Helmut Hasse]] conjectured that ''L''(''E'', ''s'') could be extended by [[analytic continuation]] to the whole complex plane. This conjecture was first proved by [[Max Deuring]] for elliptic curves with [[complex multiplication]]. It was subsequently shown to be true for all elliptic curves over '''Q''', as a consequence of the [[modularity theorem]]. Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime ''p'' is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive. == History == In the early 1960s [[Henry Peter Francis Swinnerton-Dyer|Peter Swinnerton-Dyer]] used the [[EDSAC]] computer at the [[University of Cambridge Computer Laboratory]] to calculate the number of points modulo ''p'' (denoted by ''N<sub>p</sub>'') for a large number of primes ''p'' on elliptic curves whose rank was known. From these numerical results [[Bryan Birch]] and [[Henry Peter Francis Swinnerton-Dyer|Swinnerton-Dyer]] conjectured that ''N<sub>p</sub>'' for a curve ''E'' with rank ''r'' obeys an asymptotic law :<math>\prod_{p<x} \frac{N_p}{p} \approx \log(x)^r \mbox{ as } x \rightarrow \infty. </math> Initially this was on the basis of somewhat tenuous trends in graphical plots; which induced a measure of skepticism in [[J. W. S. Cassels]] (Birch's Ph.D. advisor). Over time the numerical evidence stacked up. This in turn led them to make a general conjecture about the behaviour of a curve's L-function ''L''(''E'', ''s'') at ''s'' = 1, namely that it would have a zero of order ''r'' at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of ''L''(''E'', ''s'') there was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the [[Reciprocal (mathematics)|reciprocal]] of the L-function is from some points of view a more natural object of study; on occasion this means that one should consider poles rather than zeroes.) The conjecture was subsequently extended to include the prediction of the precise leading [[Taylor coefficient]] of the L-function at ''s'' = 1. It is conjecturally given by :<math>\frac{L^{(r)}(E,1)}{r!} = \frac{\#\mathrm{Sha}(E)\Omega_E R_E \prod_{p|N}c_p}{(\#E_{\mathrm{Tor}})^2}</math> where the quantities on the right hand side are invariants of the curve, studied by Cassels, [[John Tate|Tate]], [[Shafarevich]] and others: these include the order of the [[torsion group]], the order of the [[Tate-Shafarevich group]], and the [[canonical height]]s of a basis of rational points..<ref name=Wiles/> == Current status == The Birch and Swinnerton-Dyer conjecture has been proved only in special cases : # In 1976 [[John Coates (mathematician)|John Coates]] and [[Andrew Wiles]] proved that if ''E'' is a curve over a number field ''F'' with complex multiplication by an imaginary quadratic field ''K'' of [[class number (number theory)|class number]] 1, ''F=K'' or '''Q''', and ''L(E,1)'' is not 0 then ''E'' has only a finite number of rational points. This was extended to the case where ''F'' is any finite abelian extension of ''K'' by Nicole Arthaud-Kuhman, who shared an office with Wiles when both were students of Coates at Stanford. # In 1983 [[Benedict Gross]] and [[Don Zagier]] showed that if a [[modular elliptic curve]] has a first-order zero at ''s'' = 1 then it has a rational point of infinite order; see [[Gross–Zagier theorem]]. # In 1990 [[Victor Kolyvagin]] showed that a modular elliptic curve ''E'' for which ''L(E,1)'' is not zero has rank 0, and a modular elliptic curve ''E'' for which ''L(E,1)'' has a first-order zero at ''s'' = 1 has rank 1. # In 1991 [[Karl Rubin]] showed that for elliptic curves defined over an imaginary quadratic field ''K'' with complex multiplication by ''K'', if the ''L''-series of the elliptic curve was not zero at ''s=1'', then the ''p''-part of the Tate-Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes ''p > 7''. # In 1999 [[Andrew Wiles]], [[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]] and [[Richard Taylor (mathematician)|Richard Taylor]] proved that all elliptic curves defined over the rational numbers are modular (the [[Taniyama-Shimura theorem]]), which extends results 2 and 3 to all elliptic curves over the rationals. Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture. ==Clay Mathematics Institute Prize== 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 the first proof of the whole conjecture.<ref>[http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/ Birch and Swinnerton-Dyer Conjecture] at [[Clay Mathematics Institute]]</ref> ==Notes== {{reflist}} == External links == *{{MathWorld|urlname = Swinnerton-DyerConjecture |title = Swinnerton-Dyer Conjecture}} *{{planetmath reference|id = 4561|title = Birch and Swinnerton-Dyer Conjecture}} * [http://sums.mcgill.ca/delta-epsilon/mag/0610/mmm061024.pdf The Birch and Swinnerton-Dyer Conjecture]: An Interview with Professor [[Henri Darmon]] by Agnes F. Beaudry * [http://www.qeden.com/wiki/The_Birch_and_Swinnerton-Dyer_Conjecture QEDen] Millennium Prize Problems Wiki [[Category:Number theory]] [[Category:Zeta and L-functions]] [[Category:Diophantine geometry]] [[Category:Conjectures]] [[Category:Unsolved problems in mathematics]] [[Category:Millennium Prize Problems]] [[Category:University of Cambridge Computer Laboratory]] [[de:Vermutung von Birch und Swinnerton-Dyer]] [[es:Conjetura de Birch y Swinnerton-Dyer]] [[fr:Conjecture de Birch et Swinnerton-Dyer]] [[ko:??? ????-??? ??]] [[it:Congettura di Birch e Swinnerton-Dyer]] [[he:????? ????' ??????????-????]] [[ja:?o[?`E?X?E?B???i[?g???_?C?A[?\?z]] [[pt:Conjetura de Birch e Swinnerton-Dyer]] [[fi:Birchin ja Swinnerton-Dyerin konjektuuri]] [[zh:?q?az????-??????z]] © Martin Musatov (http://MeAmI.org) 1978-2009. All Rights Reserved. |