# On the Rank of Elliptic Curves in Elementary Cubic Extensions

Authors: Rintaro Kozuma
Journal: Journal of Numbers
 (2015)

---

## Abstract

We give a method for explicitly constructing an elementary cubic extensionL over which an elliptic curve E
D
:
y
2
+
D
y
=
x
3
(
D
∈
Q
∗
) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of E
D and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [
L
:
Q
] often becomes small.


---

## Body

## 1. Introduction

LetE be an elliptic curve defined over a number field F. It is well-known that the Mordell-Weil group E
(
F
) of F-rational points on E forms a finitely generated abelian group, and its r
a
n
k rank
⁡
E
(
F
)
=
dim
Q
⁡
E
(
F
)
⊗
Z
Q is of great interest in arithmetic geometry. The present paper is motivated by the following general problem:To understand the behavior of ranks of elliptic curves in towers of finite extensions overF.This problem dates back at least to [1], which first introduced an Iwasawa theory for elliptic curves. In the context of Iwasawa theory, Kurcanov [2] showed that if an elliptic curve E over Q without complex multiplication has good reduction at a prime number p
>
3 satisfying a mild condition then E has infinite rank in a certain Z
p-extension, and Harris [3] showed that if E is a modular elliptic curve over F having good ordinary reduction at p and a specific F-rational point arising from the modular curve then there is a p-adic Lie extension of F in which the rank of E grows infinitely. On the other hand, Kida [4] constructed a tower of elementary 2-extensions in which the rank of y
2
=
x
3
-
n
2
x becomes arbitrarily large by using a result in theory of congruent numbers. Also, Dokchitser [5] proved that for any elliptic curve E over F there are infinitely many cubic extensions K
/
F so that the rank of E increases in K
/
F. In the present paper, we consider the specific curve (1)
E
D
:
y
2
+
D
y
=
x
3
,
D
∈
Q
∗
,and give an explicit construction of an elementary cubic extension over which the elliptic curve E
D is of rank of at least a given positive integer l by finding a close connection between a 3-isogeny of E
D and a generic polynomial for cyclic cubic extensions. We shall call an extension K
/
F anelementary cubic extension if K is a finite compositum of cubic extensions over F. Compared with related results as above, in our method, the extension degree [
L
:
Q
] often becomes small. The question of finding a small elementary cubic extension with rank ≥
l seems to be of independent interest.


## 2. Main Results

Let(2)
S
D
=
b
∈
Z
∣
D
=
a
2
+
3
a
b
+
9
b
2
a
,
b
∈
Z
,
b
>
0
,
where
a
is relatively prime to
b
.We denote by F
¯ a fixed algebraic closure of F. Let F
(
S
D
3
) denote the field obtained by adjoining a real number b
3 for each b
∈
S
D to F. Since the set S
D is finite, the extension F
(
S
D
3
)
/
F is also finite. The main result of the paper is the following theorem.Theorem 1.
LetD be any square-free product of distinct primes p
1
,
…
,
p
k
∈
Z satisfying p
i
≡
1
(
m
o
d
3
). Then, for any positive integer l
(
≤
k
), we can construct a subfield L of Q
(
S
D
3
) such that (3)
rank
⁡
E
D
L
≥
l
,
L
:
Q
≤
3
l
.As we will see in Section4, the field L in Theorem 1 is effectively computable from the decomposition of the primes p
1
,
…
,
p
k in Z
[
ζ
3
], where ζ
3 denotes a primitive cube root of unity. See Remark 4 for this computation. It will also turn out in Section 4 that the field L is constructed using certain products of prime elements in Z
[
ζ
3
] above each prime p
i, and in particular L depends only on the primes p
1
,
…
,
p
k
(
≡
1
(
m
o
d
3
)
) and a positive integer l
(
≤
k
). One question arises here: how small can [
L
:
Q
] be when p
1
,
…
,
p
k vary through the set of all primes p
≡
1
(
m
o
d
3
) for a fixed integer l
(
≤
k
)? We give some examples in the case l
=
k (Table 1).Table 1
rank
⁡
E
D
(
L
)
≥
k.
D
L
rank
⁡
E
D
(
L
)
1
3
⋅
421
Q
≥2
19
⋅
277
Q
≥2
31
⋅
373
Q
≥2
109
⋅
1153
Q
≥2
7
⋅
19
⋅
37
Q
(
17
3
)
≥3
7
⋅
19
⋅
229
Q
(
3
3
)
≥3
7
⋅
43
⋅
127
Q
(
61
3
)
≥3
7
⋅
43
⋅
127
Q
(
73
3
)
≥3
109
⋅
349
⋅
457
Q
(
307
3
)
≥3Corollary 2.
LetD be any square-free product of distinct primes p
1
,
…
,
p
k
∈
Z satisfying p
i
≡
1
(
m
o
d
3
). Then, for any positive integer l, we can construct a subfield L
′ of Q
(
S
d
D
3
,
d
3
) with some integer d such that (4)
rank
⁡
E
D
L
′
≥
l
,
L
′
:
Q
≤
3
l
+
1
.Note that our construction in Theorem1 (resp., Corollary 2) differs from Dokchitser’s [5], and the extension degree [
L
:
Q
] (resp., [
L
′
:
Q
]) is often smaller than or equal to 3
l
-
1 (resp., 3
l).


## 3. A Connection between the Elliptic CurveE
D and a Generic Polynomial for Cyclic Cubic Extensions

In this section, letF be a perfect field with characteristic not equal to 3, and let D be any element in F
∗. The elliptic curve E
D has a rational 3-torsion point (
0
,
0
) and admits a 3-isogeny ϕ
:
E
D
→
E
D
^
=
E
D
/
〈
(
0
,
0
)
〉 over F. The 3-isogeny ϕ can be explicitly written as(5)
ϕ
:
E
D
⟶
E
D
^
x
,
y
⟼
x
3
+
D
2
D
x
2
,
-
-
y
/
D
3
+
3
-
y
/
D
-
1
-
y
/
D
-
y
/
D
-
1
.The isogenous elliptic curve is given by the form (6)
E
D
^
:
t
2
+
3
t
+
9
=
D
s
3
.Taking Galois cohomology of the short exact sequence 0
→
E
D
[
ϕ
]
→
E
D
→
ϕ
E
D
^
→
0 yields a Kummer map (7)
E
D
^
F
ϕ
E
D
F
↪
H
1
F
,
E
D
ϕ
P
⟼
σ
⟼
ϕ
-
1
P
σ
-
1
.Combining this map with the map (8)
H
1
F
,
E
D
ϕ
=
H
o
m
G
a
l
F
¯
F
,
Z
3
Z
↠
G
a
l
C
3
F
ξ
⟼
F
¯
K
e
r
ξ
,where G
a
l
(
C
3
/
F
) denotes the set of all cyclic cubic extensions over F together with F, (9)
G
a
l
C
3
F
≔
K
F
:
a Galois extension in
F
¯
K
F
G
a
l
K
F
↪
C
3
≃
Z
3
Z
,we have a map (10)
δ
F
:
E
D
^
F
ϕ
E
D
F
⟶
G
a
l
C
3
F
P
⟼
F
ϕ
-
1
P
.It is well-known that the setG
a
l
(
C
3
/
F
) is parametrized by a generic polynomial for C
3-extensions [6](11)
g
x
;
t
=
x
3
+
t
x
2
-
t
+
3
x
+
1
w
i
t
h
t
∈
F
,which has discriminant (
t
2
+
3
t
+
9
)
2. We shall denote by K
t the minimal splitting field of g
(
x
;
t
) over F
0
(
t
), where F
0 stands for the prime field in F. By the property of g
(
x
;
t
), for any K
/
F
∈
G
a
l
(
C
3
/
F
) there is some t
∈
F so as to be K
=
K
t
F in F
¯.The following proposition is the key observation in the present paper.Proposition 3.
The connecting homomorphismδ
F is given by δ
F
(
s
,
t
)
=
K
t
F for any (
s
,
t
)
∈
E
D
^
(
F
). Furthermore, in the case where F is a number field, for any prime ideal p of F not dividing 3, the p-component of the conductor of K
t
F
/
F is(12)
p
i
f
ν
p
D
s
3
>
0
,
ν
p
D
≢
0
m
o
d
3
1
o
t
h
e
r
w
i
s
e
.Here ν
p
:
F
p
∗
→
Z denotes the normalized valuation of the p-adic completion F
p.Proof.
For any(
s
,
t
)
∈
E
D
^
(
F
), take a point (
x
,
y
) on E
D satisfying ϕ
(
x
,
y
)
=
(
s
,
t
). By the form of the y-coordinate of (5), the splitting field K
t coincides with F
0
(
t
,
y
). Here F
(
x
,
y
)
=
F
(
x
)
=
F
(
y
). The latter part of the proposition follows from the equation t
2
+
3
t
+
9
=
D
s
3 by using Proposition  8.1 in [7].


## 4. A Method for Constructing an Elementary Cubic Extension

From now on, letD be a square-free product of distinct primes p
1
,
…
,
p
k
∈
Z satisfying p
i
≡
1
(
m
o
d
3
), and let l be a positive integer ≤
k.Recall a result of Komatsu [8] that the set of all cyclic cubic fields (i.e., cyclic cubic extensions over Q) of conductor D
=
p
1
⋯
p
k has a one-to-one correspondence to the subset of the algebraic torus T
(
Q
)
=
Q
∪
{
∞
} consisting of 2
k
-
1 elements of the form (13)
c
p
1
±
T
c
p
2
±
T
⋯
±
T
c
p
k
-
1
±
T
c
p
k
,where +
T is the composition of T
(
Q
) given by a
/
b
+
T
c
/
d
=
(
a
/
b
·
c
/
d
-
1
)
/
(
a
/
b
+
c
/
d
+
1
), -
T denotes the inverse of +
T given by -
T
a
/
b
=
-
a
/
b
-
1, and c
p
i
=
e
p
i
/
f
p
i
∈
T
(
Q
) is a rational number so that the pair (
e
p
i
,
f
p
i
) of integers is unique in the sense of Lemma  2.1 in [8] (especially f
p
i
≡
0
(
m
o
d
3
), which will be used in the following) and satisfies (14)
e
p
i
2
+
e
p
i
f
p
i
+
f
p
i
2
=
e
p
i
-
f
p
i
ζ
3
e
p
i
-
f
p
i
ζ
3
2
=
p
i
.There is a group isomorphism φ
:
T
→
G
m mapping t to (
t
-
ζ
3
)
/
(
t
-
ζ
3
-
1
), where G
m denotes the multiplicative group. Then the composition +
T is written as a
/
b
+
T
c
/
d
=
φ
-
1
(
φ
(
a
/
b
)
φ
(
c
/
d
)
).Now, letπ
i
≔
e
p
i
-
f
p
i
ζ
3, which is a prime element in Z
[
ζ
3
] dividing p
i, and let α
¯ denote the complex conjugation of α. For each i
(
1
≤
i
≤
l
), let (15)
t
i
≔
3
c
p
1
+
T
⋯
+
T
c
p
k
if
i
=
1
3
c
p
1
+
T
⋯
+
T
c
p
i
-
1
-
T
c
p
i
+
T
c
p
i
+
1
+
T
⋯
+
T
c
p
k
if
i
≠
1
.Since f
p
i
≡
0
(
m
o
d
3
) for each i, there exist unique integers a
i, b
i so that (16)
a
i
-
3
b
i
ζ
3
=
π
1
⋯
π
k
if
i
=
1
π
1
⋯
π
i
-
1
·
π
i
¯
·
π
i
+
1
⋯
π
k
if
i
≠
1
,where a
i is relatively prime to 3
b
i. Then φ
(
t
i
/
3
)
=
(
a
i
-
3
b
i
ζ
3
)
/
(
a
i
-
3
b
i
ζ
3
)
¯, and thus t
i
=
3
·
φ
-
1
(
φ
(
t
i
/
3
)
)
=
a
i
/
b
i.LetL
≔
Q
(
{
b
i
3
∣
1
≤
i
≤
l
}
).Remark 4.
In order to calculatet
i
=
a
i
/
b
i, we have only to know the values c
p
i by the definition of t
i. For example, one can use Table  5.1 in [8], in which the values c
p for p
<
1000 are listed. From the construction of L, the field L depends only on D
=
p
1
⋯
p
k and a positive integer l
(
≤
k
).Lemma 5.
Every prime numberp
i
(
1
≤
i
≤
l
) is unramified in L.Proof.
Sincep
i does not divide 3
b
1
⋯
b
l and is unramified in Q
(
ζ
3
), it turns out that p
i is unramified in Q
(
ζ
3
,
{
b
i
3
∣
1
≤
i
≤
l
}
) by Kummer theory. Hence, p
i is also unramified in L
=
Q
(
{
b
i
3
∣
1
≤
i
≤
l
}
).Proposition 6.
K
t
i
L
∈
I
m
δ
L
-
{
L
} for any i
(
1
≤
i
≤
l
).Proof.
Since(
a
i
-
3
b
i
ζ
3
)
(
a
i
-
3
b
i
ζ
3
)
¯
=
D, we have(17)
t
i
2
+
3
t
i
+
9
=
D
b
i
3
-
2
3
.Thus b
i
3
-
2
,
t
i
∈
E
D
^
(
L
). It follows from Proposition 3 that δ
L
b
i
3
-
2
,
t
i
=
K
t
i
L, and hence K
t
i
L
∈
I
m
δ
L. Combining Proposition 3 with Lemma 5 yields K
t
i
L
≠
L.Proposition 7.
d
i
m
F
3
E
D
^
(
L
)
/
ϕ
(
E
D
(
L
)
)
≥
l.Proof.
One can easily verify that(
0
,
3
ζ
3
) is a 3-torsion point in E
D
^
(
L
(
ζ
3
)
), and the rational function f
≔
(
t
-
3
ζ
3
)
/
D
2 on E
D
^ has a zero of order 3 at (
0
,
3
ζ
3
) and a pole of order 3 at infinity and satisfies (18)
f
∘
ϕ
=
y
-
D
ζ
3
D
x
3
.Then there exists an injective homomorphism (see [9] for details): (19)
E
D
^
L
ζ
3
ϕ
E
D
L
ζ
3
⟶
L
ζ
3
∗
L
ζ
3
∗
3
s
,
t
⟼
D
t
-
3
ζ
3
L
ζ
3
∗
3
0,3
ζ
3
⟼
D
2
L
ζ
3
∗
3
.Combining this with the natural map(20)
E
D
^
L
ϕ
E
D
L
⟶
E
D
^
L
ζ
3
ϕ
E
D
L
ζ
3
,we have the homomorphism(21)
E
D
^
L
ϕ
E
D
L
⟶
L
ζ
3
∗
L
ζ
3
∗
3
s
,
t
⟼
D
t
-
3
ζ
3
L
ζ
3
∗
3
.By Proposition 6, the point b
i
3
-
2
,
t
i
∈
E
D
^
(
L
) corresponds to D
(
t
i
-
3
ζ
3
)
L
(
ζ
3
)
∗
3
=
D
(
a
i
-
3
b
i
ζ
3
)
L
(
ζ
3
)
∗
3 via the above homomorphism. It thus suffices to show that {
D
(
a
i
-
3
b
i
ζ
3
)
∣
1
≤
i
≤
l
} is linearly independent in L
(
ζ
3
)
∗
/
L
(
ζ
3
)
∗
3. Then the points b
i
3
-
2
,
t
i
∣
1
≤
i
≤
l must be linearly independent in E
D
^
(
L
)
/
ϕ
(
E
D
(
L
)
). Assume that (22)
γ
≔
∏
1
≤
i
≤
l
D
a
i
-
3
b
i
ζ
3
r
i
∈
L
ζ
3
∗
3for some integers {
r
i
}
1
≤
i
≤
l. For each i, let p
i be a prime ideal of L
(
ζ
3
) above the prime element π
i
∈
Q
(
ζ
3
). For any prime ideal p of L
(
ζ
3
), we denote by ν
p
:
L
(
ζ
3
)
p
∗
→
Z the normalized (additive) valuation of the p-adic completion L
(
ζ
3
)
p. By Lemma 5, we have ν
p
i
¯
(
π
i
¯
)
=
1. Then(23)
ν
p
j
¯
γ
=
∑
i
=
1
l
r
i
≡
0
mod
⁡
3
if
j
=
1
∑
i
=
1
l
r
i
+
r
j
≡
0
mod
⁡
3
if
j
≠
1
,and hence r
i
≡
0
(
m
o
d
3
) for each i. This proves that {
D
(
a
i
-
3
b
i
ζ
3
)
∣
1
≤
i
≤
l
} is linearly independent in L
(
ζ
3
)
∗
/
L
(
ζ
3
)
∗
3.Letϕ
^
:
E
D
^
→
E
D be the dual isogeny to ϕ. Since the composition ϕ
^
∘
ϕ (resp., ϕ
∘
ϕ
^) is the multiplication-by-3 map on E
D (resp., E
D
^), we have an exact sequence of F
3-vector spaces (24)
0
⟶
E
D
^
L
ϕ
^
ϕ
E
D
L
3
⟶
E
D
^
L
ϕ
E
D
L
⟶
ϕ
^
E
D
L
3
E
D
L
⟶
E
D
L
ϕ
^
E
D
^
L
⟶
0
,and thus (25)
rank
⁡
E
D
L
=
dim
F
3
⁡
E
D
L
ϕ
^
E
D
^
L
+
dim
F
3
⁡
E
D
^
L
ϕ
E
D
L
-
dim
F
3
⁡
E
D
^
L
ϕ
^
ϕ
E
D
L
3
-
dim
F
3
⁡
E
D
L
3
.Since ζ
3
∉
L, the group E
D
^
(
L
)
[
ϕ
^
] is trivial and the group E
D
(
L
)
[
3
] is generated by the point (
0
,
0
). We see that (26)
rank
⁡
E
D
L
=
dim
F
3
⁡
E
D
L
ϕ
^
E
D
^
L
+
dim
F
3
⁡
E
D
^
L
ϕ
E
D
L
-
1
.Proof of Theorem1.
By Proposition7, it suffices to show that (
0
,
0
)
∉
ϕ
^
(
E
D
^
(
L
)
). Then (27)
dim
F
3
⁡
E
D
L
ϕ
^
E
D
^
L
≥
1
,
and  hence
rank
⁡
E
D
L
≥
l
.Assume that there exists some point Q
∈
E
D
^
(
L
) such that ϕ
^
(
Q
)
=
(
0
,
0
). Then Q
∈
E
D
^
(
L
)
[
3
]. It follows from the 3-division polynomial 3
x
(
x
3
-
27
D
2
) for E
D
^ that there is no nontrivial 3-torsion point in E
D
^
(
L
). Therefore, Q must be the trivial element in E
D
^
(
L
)
[
3
]. This contradicts the assumption, and hence (
0
,
0
)
∉
ϕ
^
(
E
D
^
(
L
)
).Proof of Corollary2.
By Theorem1, we have only to consider the case l
>
k. (In the case where l
≤
k, take d
=
1.) Take distinct prime numbers {
q
1
,
…
,
q
l
-
k
} not dividing D with q
i
≡
1
(
m
o
d
3
). Let d
≔
q
1
⋯
q
l
-
k. Then, applying Theorem 1 to the elliptic curve E
d
D, there exists some subfield L
⊂
Q
(
S
d
D
3
) so as to be rank
⁡
E
d
D
(
L
)
≥
l with [
L
:
Q
]
≤
3
l. Let L
′
≔
L
(
d
3
). Then, it is easily seen that E
d
D is isomorphic over L
′ to E
D. Therefore, rank
⁡
E
D
(
L
′
)
=
rank
⁡
E
d
D
(
L
′
)
≥
rank
⁡
E
d
D
(
L
)
≥
l. Here [
L
′
:
Q
]
≤
3
·
[
L
:
Q
]
≤
3
l
+
1.



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*Source: 501629-2015-10-08.xml*