Number Theory: An Introduction to Mathematics

572 XIII Connections with Number Theory

and|γp|=| ̄γp|=p−^1 /^2. Multiplying out the products, we obtain forRs> 3 / 2
an absolutely convergent Dirichlet series

L(s)=

∑

n≥ 1

cnn−s

with integer coefficientscn. (Ifn=pis prime, thencnis the previously definedcp.)
Theconductor N = N(W)of the elliptic curveW is defined by the singular
reductionsWpofW:

N=

∏

p|∆

pfp,

wherefp=1ifWphas a node, whereasfp=2ifp>3andWphas a cusp. We
will not definefpifp∈{ 2 , 3 }andWphas a cusp, but we mention thatfpis then an
integer≥2 which can be calculated by an algorithm due to Tate (1975). (It may be
shown thatf 2 ≤8andf 3 ≤5.)
The elliptic curveWis said to besemi-stableifWphas a node for everyp|∆. Thus,
for a semi-stable elliptic curve, the conductorNis precisely the product of the distinct
primes dividing the discriminant∆. (The semi-stable case is the only one in which the
conductor is square-free.)
Three important conjectures aboutelliptic curves, involving theirL-functions and
conductors, will now be described.
It was conjectured by Hasse (1954) that the function

ζ(s,W):=ζ(s)ζ(s− 1 )/L(s,W)

may be analytically continued to a function which is meromorphic in the whole
complex plane and thatζ( 2 −s,W)is connected withζ(s,W)by a functional
equation similar to that satisfied by the Riemann zeta-functionζ(s).Intermsof
L-functions, Hasse’s conjecture was given the following precise form by Weil (1967):

HW-Conjecture:If the elliptic curveW has L-function L(s)and conductor N , then
L(s)may be analytically continued, so that the function

Λ(s)=( 2 π)−sΓ(s)L(s),

whereΓ(s)denotes Euler’s gamma-function, is holomorphic throughout the whole
complex plane and satisfies the functional equation

Λ(s)=±N^1 −sΛ( 2 −s).

(In fact it is the functional equation which determines the precise definition of the
conductor.)

The second conjecture, due to Birch and Swinnerton-Dyer (1965), connects the
L-function with the group of rational points:

BSD-Conjecture:The L-function L(s)of the elliptic curveWhas a zero at s= 1 of
order exactly equal to the rank r≥ 0 of the group E=E(W,Q)of all rational points
ofW.