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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/ 
Associated Topics:
College Modern Algebra
College Number Theory

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