Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 439

This important conjecture combines most of the important information
about E into one equation. When it was first made, there were no exam-
ples. As Tate pointed out in 1974 ([116, p. 198]),

This remarkable conjecture relates the behavior of a function L at
a point where it is not at present known to be defined to the order
of a group which is not known to be finite!

In 1986, Rubin gave the first examples of curves with finite , and was able to
compute the exact order of in several examples. Since they were complex
multiplication curves, L E (1) could be computed explicitly by known formulas
(these had been used by Birch and Swinnerton-Dyer in their calculations), and
this allowed the conjecture to be verified for these curves. Soon thereafter,
Kolyvagin obtained similar results for elliptic curves satisfying Theorem 14.4
(which was not yet proved) such that L E ( s ) vanishes to order at most 1 at
s = 1. Therefore, the conjecture is mostly proved (up to small rational factors)
when L E ( s ) vanishes to order at most one at s = 1. In general, nothing is
known when L E ( s ) vanishes to higher order. In fact, it is not ruled out (but
most people believe it's very unlikely) that L E ( s ) could vanish at s =1to
very high order even though E ( Q )hasrank0or1.
In 2000, the Clay Mathematics Institute listed the Conjecture of Birch and
Swinnerton-Dyer as one of its million dollar problems. There are surely easier
(but certainly less satisfying) ways to earn a million dollars.
For those who know some algebraic number theory, the conjecture is very
similar to the analytic class number formula. For an imaginary quadratic field
K , the zeta function of K satisfies

2 πh
w |

ζ K ( s )=( s − 1) − 1 + ··· ,

d |

where h is the class number of K , d is the discriminant of K ,and w is the
number of roots of unity in K . Conjecture 14.7 for a curve of rank r =0
predicts that

Ω p c p #

E

L E ( s )= + ··· .

# E ( Q ) torsion

The group

E can be regarded as the an alogue of the ideal class group, the

number Ω p c p plays the role of 2 π/ |d| ,and# E ( Q ) torsion is the analogue
of w . Except for the square on the order of the torsion group, the two formulas
for the leading coecients have very similar forms.
Now let's look at real quadratic fields K . The class number formula says
that

1) − 1 4 h log( η ) ζ K ( s )=( s −

2 √ d

+ ··· ,

where h is the class number of K , d is the discriminant, and η is the fun-
damental unit. The Conjecture of Birch and Swinnerton-Dyer for a curve of