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.

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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. It uses material from the Wikipedia article " Birch and Swinnerton-Dyer conjecture ". Except where otherwise indicated, Everything.

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