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In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. It is widely recognized as one of the most challenging mathematical problems; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof.〔(Birch and Swinnerton-Dyer Conjecture ) at Clay Mathematics Institute〕 It is named after mathematicians Bryan Birch and Peter Swinnerton-Dyer who developed the conjecture during the first half of the 1960s with the help of machine computation. , only special cases of the conjecture have been proven correct. The conjecture relates arithmetic data associated to an elliptic curve ''E'' over a number field ''K'' to the behaviour of the Hasse–Weil ''L''-function ''L''(''E'', ''s'') of ''E'' at ''s'' = 1. More specifically, it is conjectured that the rank of the abelian group ''E''(''K'') of points of ''E'' is the order of the zero of ''L''(''E'', ''s'') at ''s'' = 1, and the first non-zero coefficient in the Taylor expansion of ''L''(''E'', ''s'') at ''s'' = 1 is given by more refined arithmetic data attached to ''E'' over ''K'' . == Background == proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite sub-set of the rational points on the curve, from which all further rational points may be generated. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of ''independent'' basis points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve. If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points. Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) these cannot be generalised to handle all curves. An ''L''-function ''L''(''E'', ''s'') can be defined for an elliptic curve ''E'' by constructing an Euler product from the number of points on the curve modulo each prime ''p''. This ''L''-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. It is a special case of a Hasse–Weil L-function. The natural definition of ''L''(''E'', ''s'') only converges for values of ''s'' in the complex plane with Re(''s'') > 3/2. Helmut Hasse conjectured that ''L''(''E'', ''s'') could be extended by analytic continuation to the whole complex plane. This conjecture was first proved by for elliptic curves with complex multiplication. It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem. Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime ''p'' is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Birch and Swinnerton-Dyer conjecture」の詳細全文を読む スポンサード リンク
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