Cryptography Reference
In-Depth Information
to be the image of
Z
[
T
2
,T
3
,T
5
,...
] in the endomorphism ring of
S
(
N
) (the
endomorphism ring of
S
(
N
) is the ring of linear transformations from the
vector space
S
(
N
) to itself).
A
normalized eigenform
of level
N
is a newform
f
=
∞
b
n
q
n
∈
S
(
N
)
n
=1
of level
N
with
b
1
= 1 and such that
T
r
(
f
)=
b
r
f
for all
r.
It can be shown that the space of newforms in
S
(
N
) has a basis of normalized
eigenforms. Henceforth, essentially all of the modular forms that we encounter
will be normalized eigenforms of level
N
. Often, we shall refer to them simply
as modular forms.
Let
f
be a normalized eigenform and suppose the coe
cients
b
n
of
f
are
rational integers. In this case, Eichler and Shimura showed that
f
determines
an elliptic curve
E
f
over
Q
,and
E
f
has the property that
b
r
=
a
r
for all
r
(where
a
r
=
r
+1
−
#
E
f
(
F
r
) for the primes of good reduction).
In particular, the potential modular form
f
E
f
for
E
is the modular form
f
.
Moreover,
E
f
has good reduction at the primes not dividing
N
. This result
is, in a sense, a converse of the conjecture of Taniyama-Shimura-Weil. The
conjecture can be restated as claiming that every elliptic curve
E
over
Q
arises from this construction. Actually, we have to modify this statement a
little. Two elliptic curves
E
1
and
E
2
are call
ed
isogenous
over
Q
if there is
a nonconstant homomorphism
E
1
(
Q
)
→ E
2
(
Q
) that is described by rational
functions over
Q
(see Chapter 12). It can be shown that, in this case,
f
E
1
=
f
E
2
. Conversely, Faltings showed that if
f
E
1
=
f
E
2
then
E
1
and
E
2
are
isogenous. Since only one of
E
1
,E
2
can be the curve
E
f
, we must ask whether
an elliptic curve
E
over
Q
is isogenous to one produced by the result of Eichler
and Shimura. Theorem 14.4 says that the answer is yes.
If we have an elliptic curve
E
, how can we predict what
N
should be? The
smallest possible
N
is called the
conductor
of
E
.For
E
=
E
f
, the primes
dividing the conductor
N
are exactly the primes of bad reduction of
E
f
(these
are also the primes of bad reduction of any curve isogenous to
E
f
over
Q
).
Moreover,
p
N
and
p
2
N
if and only if
E
f
has multiplicative reduction at
p
.
Therefore, if
E
f
is semistable, then
N
=
p
|
p,
(15.7)
|
Δ
namely, the product of the primes dividing the minimal discriminant Δ. We
see that
N
is squarefree if and only if
E
f
is semistable. Therefore, if
E
is an
arbitrary modular semistable elliptic curve over
Q
,then
N
is given by (15.7).
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