Helmut Hasse conjectured that 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, as a consequence of the Taniyama-Shimura theorem. Finding rational points on a general elliptic curve is a difficult problem.

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Helmut Hasse conjectured that 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, as a consequence of the Taniyama-Shimura theorem. Finding rational points on a general elliptic curve is a difficult problem.

Finding the points on an elliptic curve modulo a given prime 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 s Peter Swinnerton-Dyer used the EDSAC computer at the University of Cambridge Computer Laboratory to calculate the number of points modulo denoted by for a large number of primes on elliptic curves whose rank was known.

From these numerical results Bryan Birch and Swinnerton-Dyer conjectured that for a curve with rank obeys an asymptotic law Initially this was on the basis of somewhat tenuous trends in graphical plots; which induced a measure of scepticism in J.

Over time the numerical evidence stacked up. This was a far-sighted conjecture for the time, given that the analytic continuation of there was only established for curves with complex multiplication, which were also the main source of numerical examples. NB that the reciprocal of the -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 -function at.

It is conjecturally given by a complex formula involving invariants of the curve, studied by Cassels, Tate, Shafarevich and others: these include the order of the torsion group, the order of the Tate-Shafarevich group, and the canonical heights of a basis of rational points.

In John Coates and Andrew Wiles proved that if E is a curve with complex multiplication and is not 0 then has only a finite number of rational points, in the case of class number 1. This was extended to all imaginary quadratic fields by Nicole Arthaud. In Benedict Gross and Don Zagier showed that if a modular elliptic curve has a first-order zero at then it has a rational point of infinite order; see Gross-Zagier theorem.

In Victor Kolyvagin showed that a modular elliptic curve for which is not zero has rank 0, and a modular elliptic curve for which has a first-order zero at has rank 1. In Karl Rubin showed that for elliptic curves defined over an imaginary quadratic field with complex multiplication by -series of the elliptic curve was not zero at , then the -part of the Tate-Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes.

In Andrew Wiles, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor proved that all elliptic curves defined over the rational numbers are modular the Taniyama-Shimura theorem , which extends the previous two results 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.

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## Conjecture de Birch et Swinnerton-Dyer

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.

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## Birch and Swinnerton-Dyer Conjecture

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.

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## Vermutung von Birch und Swinnerton-Dyer

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.