The Birch and Swinnerton-Dyer Conjecture

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We let Delta denote the discriminant of the cubic and set N_p := #{solutions of y^2 = x^3 + ax + b mod p}, a_p := p - N_p.

en · official · sha256:230e7c0bd6512d04666ef177dbd0a55886f8935c41a32e6869d47a84370e04c0EN

The first general result proved was for elliptic curves with complex multiplication. This theorem was proved in 1976 and is due to Coates and Wiles. It states that if C is a curve with complex multiplication and L(C,1) != 0, then C(Q) is finite.

Ссылаются нормы: names-coates-wiles-1976

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One very old problem concerned with rational points on elliptic curves is the congruent number problem: which rational integers occur as the areas of right-angled triangles with rational length sides. Such integers are called congruent numbers. It is closely related to the problem of determining the rational points on the curve C_n: y^2 = x^3 - n^2*x. Indeed, C_n(Q) is infinite if and only if n is a congruent number.

Ссылаются нормы: defines-congruent-number

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Conjecture (Birch and Swinnerton-Dyer). The Taylor expansion of L(C,s) at s=1 has the form L(C,s) = c*(s-1)^r + higher order terms with c != 0 and r = rank(C(Q)).

Ссылаются нормы: BsdBasicFormalStatement, basic-identifies-bsd, source-asserts-basic

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In particular this conjecture asserts that L(C,1) = 0 if and only if C(Q) is infinite.

Ссылаются нормы: BsdBasicEquivalentStatement, source-asserts-basic-equivalent

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There is a refined version of this conjecture. In this version one has to define Euler factors at primes p | 2*Delta to obtain the completed L-series, L*(C,s). The conjecture then predicts that L*(C,s) ~ c*(s-1)^r with c* = |Sha_C|*R_infinity*w_infinity*(product over p|2Delta of w_p) / |C(Q)^tors|^2.

Ссылаются нормы: BsdRefinedFormalStatement, defines-sha, refined-identifies-bsd, source-asserts-refined

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In 1983 Gross and Zagier showed that if C is a modular elliptic curve and L(C,1) = 0 but L'(C,1) != 0, then an earlier construction of Heegner actually gives a rational point of infinite order. Using new ideas together with this result, Kolyvagin showed in 1990 that for modular elliptic curves, if L(C,1) != 0 then r = 0 and if L(C,1) = 0 but L'(C,1) != 0 then r = 1. In the former case Kolyvagin needed an analytic hypothesis which was confirmed soon afterwards.

Ссылаются нормы: names-gross-zagier, names-kolyvagin-1990

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Then we can define the incomplete L-series of C (incomplete because we omit the Euler factors for primes p | 2*Delta) by L(C,s) := product over p not dividing 2*Delta of (1 - a_p*p^(-s) + p^(1-2s))^(-1).

Ссылаются нормы: defines-l-series

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It is now known that all elliptic curves over Q are modular ([Wiles 1995], [Taylor-Wiles 1995], [Breuil-Conrad-Diamond-Taylor 2001]).

Ссылаются нормы: names-modularity

en · official · sha256:098c9fb15688896e9744f50aed24666ac03f91def5672798dd0d46cedba4adf9EN

Theorem. If C is an elliptic curve over Q, then C(Q) = Z^r (+) C(Q)^tors for some integer r >= 0, where C(Q)^tors is a finite abelian group. The integer r is called the rank of C. It is zero if and only if C(Q) is finite.

Ссылаются нормы: defines-rank

en · official · sha256:41c1bd6d5174d90c6798a8f9fe95634012c0194a3978500db9e797cdc5b19c25EN

Theorem. If L(C,s) ~ c*(s-1)^m with c != 0 and m = 0 or 1, then the conjecture holds.

Ссылаются нормы: defines-partial-results

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Here |Sha_C| is the order of the Tate-Shafarevich group of the elliptic curve C, a group which is not known in general to be finite although it is conjectured to be so. It counts the number of equivalence classes of homogeneous spaces of C which have points in all local fields. The term R_infinity is an r x r determinant whose matrix entries are given by a height pairing applied to a system of generators of C(Q)/C(Q)^tors. The w_p's are elementary local factors and w_infinity is a simple multiple of the real period of C.

en · official · sha256:64e3a681748edc0c630a374a94007cb90c80451d4843076264dc3d3f2e3b9578EN

THE BIRCH AND SWINNERTON-DYER CONJECTURE ANDREW WILES

Ссылаются нормы: CmiBsdSource, basic-classified, classified_as_millennium_problem, defines_notion, names_historical_result

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Assuming the Birch and Swinnerton-Dyer conjecture (or even the weaker statement that C_n(Q) is infinite iff L(C_n,1) = 0) one can show that any n congruent to 5, 6 or 7 mod 8 is a congruent number, and, moreover, Tunnell has shown, again assuming the conjecture, that for n odd and square-free: n is a congruent number if and only if #{x,y,z in Z : 2x^2 + y^2 + 8z^2 = n} = 2 * #{x,y,z in Z : 2x^2 + y^2 + 32z^2 = n}, with a similar criterion if n is even. Tunnell proved the implication left to right unconditionally with the help of the main theorem of Coates-Wiles [1977].

Ссылаются нормы: names-tunnell-1983

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We can find an affine model for the curve in Weierstrass form C: y^2 = x^3 + ax + b with a, b in Z.

Ссылаются нормы: defines-weierstrass