Page 317 -

292 15. Elliptic Curves over Global Fields and -Adic Representations

for E with coefficients in R (v) having discriminant v , differential ω r and conductor

f v . The discriminant v has the minimal property: v( v ) ≤ v( ), where is the
discriminant of any other equation for E over R (v) .
(1.1) Definition. With the above notations for an elliptic curve E over K,wede-

fine two divisors on R: D E = v( v )v, called the discriminant of E,and
v

f E = v( f v )v, called the conductor of E. When R is a principal ideal ring, for
v
example Z, these divisors are principal: D E = (d E ) and f E = (N E ), where d E in R
is the discriminant of E and N E = N in R is the conductor. These elements of R are
well-defined up to a unit in R, and in the case of the integers R = Z, we choose the
conductor always to be positive.
The conductor N E can only be defined from local data, while for the discriminant
d E , we have a discriminant F for each equation F of E over R
3
2
2
F(x, y) = y + a 1 xy + a 3 y − x − a 2 x − a 4 x − a 6 = 0
with which we can compare d E . The curve E given by equation F = 0 also has an
invariant differential ω F . Recall that, if F = 0 is a second normal cubic equation

12
for E over R, then for some nonzero u we have u F = F and u −1 ω F = ω F .

From the relation u −1 ω F = ω F , we can define the valuation of the quotient of two

differentials by

ω F −1
v = v(u ) =−v(u).
ω F
This is related to the valuations of the corresponding discriminants by
12v(ω F /ω F ) = v( F ) − v( F ).

There is an obvious transitivity condition for three curves F = 0, F = 0, and

F = 0

ω F ω F ω F
v + v = v .
ω F ω F ω F
For ω F and each local ω v , we form a divisor which relates F = 0 to the local minimal
models

ω F

A F = v v.
ω v
v
This satisfies A F + (u) = A F from u −1 ω F = ω F and the transitivity formula.

Moreover, this and the relation

ω F
12v = v( v ) − v( F )
ω v
give the following result.