Current Status
The Birch and Swinnerton-Dyer conjecture has been proved only in special cases:
- Coates & Wiles (1977) proved that if E is a curve over a number field F with complex multiplication by an imaginary quadratic field K of class number 1, F = K or Q, and L(E, 1) is not 0 then E(F) is a finite group. This was extended to the case where F is any finite abelian extension of K by Arthaud (1978).
- Gross & Zagier (1986) showed that if a modular elliptic curve has a first-order zero at s = 1 then it has a rational point of infinite order; see Gross–Zagier theorem.
- Kolyvagin (1989) showed that a modular elliptic curve E for which L(E, 1) is not zero has rank 0, and a modular elliptic curve E for which L(E, 1) has a first-order zero at s = 1 has rank 1.
- Rubin (1991) showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the L-series of the elliptic curve was not zero at s = 1, then the p-part of the Tate–Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7.
- Breuil et al. (2001), extending work of Wiles, proved that all elliptic curves defined over the rational numbers are modular, which extends results 2 and 3 to all elliptic curves over the rationals, and shows that the L-functions of all elliptic curves over Q are defined at s = 1.
- Bhargava & Shankar (2010) proved that the average rank of the Mordell–Weil group of an elliptic curve over Q is bounded above by 7/6. Combining this with the p-parity theorem by Dokchitser & Dokchitser (2010) and the announced proof of the main conjecture of Iwasawa theory for GL(2) by Skinner & Urban (2010), they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by Kolyvagin (1989), 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.
Read more about this topic: Birch And Swinnerton-Dyer Conjecture
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