Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 438

If E hasalotofpointsmod p for many p , then many factors in the product are
small, so we expect that L E (1) might be small. In fact, the data that Birch
and Swinnerton-Dyer obtained led them to make the following conjecture.

CONJECTURE 14.6 (Conjecture of Birch and Swinnerton-Dyer,
Weak Form)
Let E be an elliptic curve defined over Q . T he order of vanishing of L E ( s )
at s =1 isthe rank r of E ( Q ) .Inother w ords, if E ( Q ) torsion ⊕ Z r ,then
L E ( s )=( s − 1) r g ( s ) ,with g (1) =0 ,∞ .

One consequence of the conjecture is that E ( Q ) is infinite if and only if
L E (1) = 0. This statement remains unproved, although there has been some
progress. In 1977, Coates and Wiles showed that if E has complex multipli-
cation and has a point of infinite order, then L E (1) = 0. The results of Gross
and Zagier on Heegner points (1983) imply that if E is an elliptic curve over
Q such that L E ( s ) vanishes to order exactly 1 at s = 1, then there is a point
of infinite order. However, if L E ( s ) vanishes to order higher than 1, nothing
has been proved, even though there is conjecturally an abundance of points
of infinite order. This is a common situation in mathematics. It seems that a
solution is often easier to find when it is essentially unique than when there
are many choices.
Soon, Conjecture 14.6 was refined to give not only the order of vanishing,
but also the leading coecient of the expansion at s = 1. To state the
conjecture, we need to introduce some notation. If P 1 ,...,P r form a basis for
thefreepartof E ( Q ), then

E ( Q )= E ( Q ) torsion ⊕ Z P 1 ⊕···⊕ Z P r .

Recall the height pairing P, Q defined in Section 8.5. We can form the r × r
matrix and compute its determinant to obtain what is known as the

elliptic regulator for E .If r = 0, define this determinant to equal 1. Let
ω 1 ,ω 2 be a basis of a lattice in C that corresponds to E by Theorem 9.21.
We may assume that ω 2 ∈ R , by Exercise 9.5. If E [2]

P i ,P j

E ( R ), let Ω = 2 ω 2 .
Otherwise, let Ω = ω 2 . For each prime p , there are integers c p that we won't
define, except to say that if p is a prime of good reduction then c p =1.
A formula for computing them is given [117]. Finally, recall that

⊂

is the

(conjecturally finite) Shafarevich-Tate group of E .

CONJECTURE 14.7 (Conjecture of Birch and Swinnerton-Dyer)
Let E be an elliptic curve defined over Q .Let r be the rank of E ( Q ) .Then
L E ( s )=( s − 1) r Ω p c p (# E )det P i ,P j +( s − 1) r +1 ( b r +1 + ··· ) .

# E ( Q ) torsion