Then, for every
m
∈
Z
, we have (from (b))
ˆ
h
([
m
]
P
) =
m

2
ˆ
h
(
P
) = 0
. Now (e)
implies that there exists a constant
C
such that for each
m
∈
Z
, we have
h
f
([
m
]
P
) =

deg(
f
)
ˆ
h
([
m
]
P
)

h
f
([
m
]
P
)
 ≤
C.
So
{
P,
2
P,
3
P, . . .
} ⊆ {
Q
∈
E
(
K
)

h
f
(
Q
)
≤
C
}
, and therefore
P
has finite
order since this last set is finite.
(f) Suppose that
ˆ
h
satisfies
ˆ
h
◦
[
m
] =
m
2
ˆ
h
and
(deg
f
)
ˆ
h
=
h
f
+
O
(1)
for some
m
≥
2
. Then
ˆ
h
◦
[
m
N
] =
m
2
N
ˆ
h
and
ˆ
h
=
m

2
N
ˆ
h
◦
[
m
N
]
=
m

2
N
(
ˆ
h
◦
[
m
N
] +
O
(1))
=
ˆ
h
+
m

2
N
O
(1)
(since
ˆ
h
satisfies (b)). Now let
N
→ ∞
to obtain
ˆ
h
=
ˆ
h
.
Lemma 8.29
Suppose that
V
is a finitedimensional
R
vector space, and let
L
⊂
V
be a lattice. Let
q
:
V
→
R
be a positive definite quadratic form satisfying:
(i) If
P
∈
L
, then
q
(
P
) = 0
iff
P
= 0
.
(ii) For every constant
C
,
#
{
P
∈
L

q
(
P
)
≤
C
}
<
∞
. Then
q
is positive definite
on
V
.
Proof.
We may choose a basis of
V
such that for any
X
= (
x
1
, . . . , x
n
)
∈
V
, we have
q
(
X
) =
s
i
=1
x
2
i

t
i
=1
x
2
s
+
i
,
where
s
+
t
≤
n
= dim(
V
)
. We may view
V
R
n
via this choice of basis. Suppose
that
s
=
n
. Let
λ
be the length of the shortest vector in
L
, i.e.
λ
= inf
{
q
(
P
)

P
∈
L, P
= 0
}
.
Then (i) and (ii) imply that
λ >
0
. Now consider the set
B
(
δ
) :=
(
x
1
, . . . , x
n
)
∈
R
n
x
2
1
+
· · ·
+
x
2
s
≤
λ
2
, x
2
s
+1
+
· · ·
+
x
2
t
≤
δ
.
107
Then length (using
q
!) of any vector in
B
(
δ
)
is at most
λ/
2
, and so
B
(
δ
)
∩
L
=
{
0
}
.
Now
B
(
δ
)
is compact, convex, and symmetric about the origin, and
Vol(
B
(
δ
))
→ ∞
as
δ
→ ∞
. This contradicts Minkowski’s convex body theorem.
Theorem 8.30
(Minkowski.)
Let
L
be a lattice in
R
n
with fundamental paral
lelepiped
D
, and suppose that
B
⊆
R
n
is compact, convex, and symmetric about the
origin. If
Vol(
B
)
≥
2
n
Vol(
D
)
, then
B
contains a nonzero point of
L
.
Proof.
We claim that if
S
is a measurable set in
R
n
with
Vol(
S
)
>
Vol(
D
)
, then
S
contains distinct points
α
and
β
with
α, β
∈
L
.
Note that
Vol(
S
) =
∈
L
Vol(
S
∩
(
D
+ ))
,
D
will contain a unique translate (by an element of
L
) of each set
S
∩
(
D
+ )
. Since
Vol(
S
)
>
Vol(
D
)
, at least two of these sets will overlap, so there exist
α, β
∈
S
such
that
α

λ
=
β

λ
for distinct
λ, λ
∈
L
, so
α

β
=
λ

λ
∈
L
\ {
0
}
, as claimed.
Now take
S
=
1
2
B
=
x
2
x
∈
B
. Then
Vol(
S
) =
1
2
n
Vol(
B
)
>
Vol(
D
)
, so there exist
α, β
∈
B
such that
α
2

β
2
∈
L
. Since
B
is symmetric about the origin,

β
∈
B
. Since
B
is convex,
1
2
(
α
+ (

β
))
∈
B
.
Theorem 8.31
The NéronTate height is a positive definite quadratic form on
R
⊗
E
(
K
)
.
Proof.
Apply Lemma 8.29 to the lattice
E
(
K
)
/E
(
K
)
tors
in
E
(
K
)
⊗
R
.
Definition 8.32
The
NéronTate height pairing
on
E/K
is defined by
,
:
E
(
¯
K
)
×
E
(
¯
K
)
→
R
, given by
P, Q
=
ˆ
h
(
P
+
Q
)

ˆ
h
(
P
)

ˆ
h
(
Q
)
.
Definition 8.33
The
elliptic regulator
R
E/K
is the volume of the fundamental
108
domain of
E
(
K
)
/E
(
K
)
tors
with respect to
ˆ
h
, i.e. if
P
1
, . . . , P
r
∈
E
(
K
)
form a basis of
E
(
K
)
/E
(
K
)
tors
, then
R
E/K
:= det(
P
i
, P
j
)
. (If
r
= 0
, set
R
E/K
= 1
.)
Corollary 8.34
R
E/K
>
0
.
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 Fall '13
 SimonRubinsteinSalzedo
 Math, Algebra, Number Theory, Integral domain, Algebraic geometry, deg, Elliptic curve cryptography, Algebraic curve, elliptic curves