Jump to navigation Jump to search In mathematics , the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. This mathematics-related article is a stub. You can help Wikiquote by expanding it. Quotes[ edit ] The BSD Conjecture has its natural context within the larger scope of modern algebraic geometry and number theory.

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This was extended to the case where F is any finite abelian extension of K by. Combining this with the p-parity theorem of and and with the proof of the main conjecture of Iwasawa theory for GL 2 by, they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by, satisfy the Birch and Swinnerton-Dyer conjecture.

Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture. Assuming the Birch and Swinnerton-Dyer conjecture, is the area of a right triangle with rational side lengths a congruent number if and only if the number of triplets of integers satisfying is twice the number of triplets satisfying.

The interest in this statement is that the condition is easily verified. Admitting the BSD conjecture, these estimations correspond to information about the rank of families of elliptic curves in question. For example: suppose the generalized Riemann hypothesis and the BSD conjecture, the average rank of curves given by is smaller than.

Manjul Bhargava. Arul Shankar. Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0. Bryan John Birch. Peter Swinnerton-Dyer. Notes on Elliptic Curves II. Christophe Breuil. Brian Conrad. Fred Diamond. Richard Taylor mathematician. Book: J. John Coates mathematician. Kenneth Alan Ribet. Karl Rubin. Arithmetic Theory of Elliptic Curves.

Lecture Notes in Mathematics. Andrew Wiles. On the conjecture of Birch and Swinnerton-Dyer. Max Deuring.

Tim Dokchitser. Vladimir Dokchitser. On the Birchâ€”Swinnerton-Dyer quotients modulo squares. Benedict H. Benedict Gross. Don B. Don Zagier. Heegner points and derivatives of L-series. Victor Kolyvagin. USSR Izv. Louis Mordell. On the rational solutions of the indeterminate equations of the third and fourth degrees. On the parity of ranks of Selmer groups IV. Christopher Skinner. The Iwasawa main conjectures for GL2. Jerrold B.

Second Series. Encyclopedia: Wiles. The Birch and Swinnerton-Dyer conjecture. Arthur Jaffe. The Millennium prize problems. American Mathematical Society. Book: Koblitz, Neal. Neal Koblitz. Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Roger Heath-Brown. Duke Mathematical Journal.

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## Birch and Swinnerton-Dyer Conjecture

Over the coming weeks, each of these problems will be illuminated by experts from the Australian Mathematical Sciences Institute AMSI member institutions. Elliptic curves have a long and distinguished history that can be traced back to antiquity. They are prevalent in many branches of modern mathematics, foremost of which is number theory. The reason for this historical confusion is that these curves have a strong connection to elliptic integrals , which arise when describing the motion of planetary bodies in space. The ancient Greek mathematician Diophantus is considered by many to be the father of algebra. His major mathematical work was written up in the tome Arithmetica which was essentially a school textbook for geniuses.

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## What is the Birch and Swinnerton-Dyer conjecture?

Its zeta function is where. Analogous to the Euler factors of the Riemann zeta function, we define the local -factor of When evaluating its value at , we retrieve the arithmetic information at , Notice that each point in reduces to a point in. So when tends to be small. Birch and Swinnerton-Dyer did numerical experiments and suggested the heuristic The is defined to be the product of all local -factors, Formally evaluating the value at gives So intuitively the rank of will correspond to the value of at 1: the larger is. However, the value of at does not make sense since the product of only converges when can be continued to an analytic function on the whole of , it may be reasonable to believe that the behavior of at contains the arithmetic information of the rank of.