Number Theory: An Introduction to Mathematics

5 Further Results and Conjectures 573

This is sometimes called the ‘weak’ conjecture of Birch and Swinnerton-Dyer,
since they also gave a ‘strong’ version, in which the nonzero constantCsuch that

L(s)∼C(s− 1 )r fors→ 1

is expressed by other arithmetic invariants ofW. The strong conjecture may be
regarded as an analogue for elliptic curves ofa known formula for the Dedekind zeta-
function of an algebraic number field. An interesting reformulation of the strong form
has been given by Bloch (1980).
The statement of the third conjecture requires some preparation. For any positive
integerN,letΓ 0 (N)denote the multiplicative group of all matrices

A=

(

ab

cd

)

,

wherea,b,c,dare integers such thatad−bc=1andc≡0modN. A functionf(τ)
which is holomorphic forτ∈H (the upper half-plane) is said to be amodular form
of weight 2 forΓ 0 (N)if, for every suchA,

f((aτ+b)/(cτ+d))=(cτ+d)^2 f(τ).

An elliptic curveW, withL-function

L(s)=

∑

n≥ 1

cnn−s

and conductorN,issaidtobemodularif the function

f(τ)=

∑

n≥ 1

cne^2 πinτ,

which is certainly holomorphic inH, is a modular form of weight 2 forΓ 0 (N).This
actually implies thatfis a ‘cusp form’ and satisfies a functional equation

f(− 1 /Nτ)=∓Nτ^2 f(τ).

It follows that theMellin transform

Λ(s)=

∫∞

0

f(iy)ys−^1 dy

may be analytically continued for alls∈Cand satisfies the functional equation

Λ(s)=±N^1 −sΛ( 2 −s).

(Note the reversal of sign.) But

Λ(s)=( 2 π)−sΓ(s)L(s),

since, by (9) of Chapter IX,
∫∞

0

e−^2 πnyys−^1 dy=( 2 πn)−sΓ(s).

Hence any modular elliptic curve satisfies theHW-conjecture.
It was shown by Weil (1967) that, conversely, an elliptic curve is modular if
not only itsL-functionL(s) =

∑

n≥ 1 cnn

−s has the properties required in the