An excellent reference for this section is Andrew Wiles's Clay Math Institute paper [Wil00]. The reader is also strongly encouraged to look Birch's original paper [Bir71] to get a better sense of the excitement surrounding this conjecture, as exemplified in the following quote:
``I want to describe some computations undertaken by myself and Swinnerton-Dyer on EDSAC by which we have calculated the zeta-functions of certain elliptic curves. As a result of these computations we have found an analogue for an elliptic curve of the Tamagawa number of an algebraic group; and conjectures (due to ourselves, due to Tate, and due to others) have proliferated.''
An elliptic curve over a field
is the projective closure
of the zero locus of a nonsingular affine curve
An elliptic curve has genus , and the set of points on has
a natural structure of abelian group, with identity element the one
extra projective point at . Again, there are simple algebraic
formulas that, given two points and on an elliptic curve,
produce a third point on the elliptic curve. Moreover,
if and both have coordinates in , then so does .
The Mordell-Weil group
In the 1920s, Mordell proved that if , then is finitely
generated, and soon after Weil proved that is finitely
generated for any number field , so
Fix an elliptic curve over . For all but finitely many prime numbers , the equation (1.1.1) reduces modulo to define an elliptic curve over the finite field . The primes that must be excluded are exactly the primes that divide the discriminant of (1.1.1).
As above, the set
of points
is an abelian group. This group
is finite, because it is contained in the set
of
rational points in the projective plane. Moreover,
since it is the set of points on a (genus 1) curve,
a theorem of Hasse implies that
A deep theorem of Wiles et al. [Wil95,BCDT01], which many consider the crowning
achievement of 1990s number theory, implies that
can be analytically continued to an analytic function
on all . This implies that
has a Taylor series
expansion about :
This problem is extremely difficult. The conjecture was made in the 1960s, and hundreds of people have thought about it for over 4 decades. The work of Wiles et al. on modularity in late 1999, combined with earlier work of Gross, Zagier, and Kolyvagin, and many others proves the following partial result toward the conjecture.
In 2000, Conjecture 1.1 was declared a million dollar millenium prize problem by the Clay Mathematics Institute, which motivated even more work, conferences, etc., on the conjecture. Since then, to the best of my knowledge, not a single new result directly about Conjecture 1.1 has been proved. The class of curves for which we know the conjecture is still the set of curves over with , along with a finite set of individual curves on which further computer calculations have been performed (by Cremona, Watkins, myself, and others).
``A new idea is needed.''
- Nick Katz on BSD, at a 2001 Arizona Winter School
And another quote from Bertolini-Darmon (2001):
``The following question stands as the ultimate challenge concerning the Birch and Swinnerton-Dyer conjecture for elliptic curves over : Provide evidence for the Birch and Swinnerton-Dyer conjecture in cases where .''
William 2007-05-25