The Birch and Swinnerton-Dyer Conjecture
Date: 06/20/2012 at 02:43:35 From: pavel Subject: can you explain what is actually 'Birch and Swinnerton-Dyer Conjecture' Can you explain to me what the "Birch and Swinnerton-Dyer Conjecture" is? It's an unsolved Millennium Problem on the Clay Mathematics Institute's website, which begins describing the conjecture this way: Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x, y, z to algebraic equations like x^2 + y^2 = z^2. Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult.... What is the more complicated equation they're referring to?
Date: 06/21/2012 at 20:53:02 From: Doctor Vogler Subject: Re: can you explain what is actually 'Birch and Swinnerton-Dyer Conjecture' Hi Pavel, Thanks for writing to Dr. Math. Wikipedia gives a good introduction to the "B and SD" conjecture: http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture Essentially, it's a statement about abelian varieties: http://en.wikipedia.org/wiki/Abelian_variety In particular, it's about the simplest type of abelian variety, which is elliptic curves: http://en.wikipedia.org/wiki/Elliptic_curve B and SD relates the structure of the algebraic group of rational points on that curve to an associated L-function: http://en.wikipedia.org/wiki/L-function http://en.wikipedia.org/wiki/Hasse-Weil_L-function This is a complicated function related to the elliptic curve in a way that defies easy description. But here's an example. One elliptic curve is given by the equation: y^2 = x^3 - x + 1 This curve has rank 1, which means, in particular, that there are infinitely many solutions where x and y are rational numbers: [1, 1] [-1, 1] [0, -1] [3, -5] [5, 11] [1/4, 7/8] [-11/9, -17/27] [19/25, -103/125] [56, -419] [159/121, 1861/1331] [-255/361, 7981/6859] [-223/784, -24655/21952] [5665/2809, -399083/148877] [26239/2601, 4231459/132651] [23464/49729, 8824453/11089567] [-350701/265225, -13919407/136590875] [1044021/1907161, -2068194649/2633789341] [9840321/702244, -30795303833/588480472] [78094085/43784689, 640700244397/289723287113] [-164914877/419061841, 9902960463475/8578614947111] [-1157009840/1947192129, -101098481076377/85923747076383] [33533414239/22994386321, -5668823512883159/3486845747330119] ... and so on. It is not easy to determine the rank of an elliptic curve, although there are algorithms that will do this. They can take a long time when the coefficients of the curve are large, and they might be even harder for abelian varieties that are not elliptic curves. The B and SD conjecture says that this rank can be determined from an understanding of the L-function of this curve. Note that finding rational solutions to this curve ... y^2 = x^3 - x + 1 ... is equivalent to finding integer solutions to this curve: z*y^2 = x^3 - x*z^2 + z^3. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
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