Download Torsion Groups of Elliptic Curves with Everywhere Good

International Journal of Algebra, Vol. 10, 2016, no. 10, 461 - 467
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ija.2016.6861
Torsion Groups of Elliptic Curves with Everywhere
Good Reduction over Quadratic Fields
Takaaki Kagawa
Department of Mathematical Sciences
Ritsumeikan University
1-1-1, Nojihigashi, Kusatsu, Shiga, 525–8577, Japan
c 2016 Takaaki Kagawa. This article is distributed under the Creative ComCopyright mons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Abstract
This paper concerns elliptic curves defined over quadratic fields with
everywhere good reduction. The curves are classified over the structure
of the torsion subgroup. There are only 5 torsion types and 22 curves
if the torsion subgroups are not isomorphic to {0}, Z/2Z and Z/3Z.
Mathematics Subject Classification: 11G05
Keywords: Ellitpic curves with everywhere good reduction, Torsion subgroups
1
Introduction
We first recall terminology “admissible elliptic curve” defined over a number
field K. If an elliptic curve defined over K with everywhere good reduction has
a K-rational 2-division point, then it is said to be admissible. Comalada [1]
studies the admissible curves and proves many important results. On the other
hand, Zhao [19] gives terminology 3-admissible. An elliptic curve defined over
a number field K is called 3-admissible if it has everywhere good reduction over
K and has a K-rational 3-division point. He proves that there are infinitely
many 3-admissible curves in the case where K is real quadratic.
Here we introduce a new notion. Let E be an elliptic curve over a number
field K. For a finite group G, E is called G-admissible if it has everywhere
462
Takaaki Kagawa
good reduction over K and E(K)tors ∼
= G. (Note that Comalada uses the
word “g-admissible”. But his “g” means “allowing global minimal equation”.
So our terminology is not the cause of confusion.)
When G ∼
= Z/2Z or G ∼
= Z/3Z, the number of G-admissible curves over
real quadratic field is infinite, as is referred above. In this paper, we determine
all G-admissible curves over quadratic fields when G 6∼
= {0}, Z/2Z and Z/3Z.
Remark. For an entire collection of the torsion group structures of elliptic
curves over quadratic fields, see [7].
2
Result
We introduce our main theorem.
Theorem 2.1. Let K be a quadratic field. Let G be a finite group not
isomorphic to {0}, Z/2Z and Z/3Z, and let E√be a G-admissible
curve
√
√ elliptic √
defined
√ over K.
√ Then K must be either Q( 6), Q( 7), Q( 26), Q( 37),
Q( 41) or Q( 65), and E is isomorphic to a curve in Table 1 through Table
5.
We explain the contents of the tables.
(1) The code, what the author uses in his thesis √
[4], is of the form mXi,
where m indicates that the curve is defined over Q( m), X = A, B denotes
a K-isogeny class (but in our tables, except for m = 65, only 1 isogeny√class
exists over an each field), and i is the ordinal number of the curve over Q( m).
(When m = 6, 17, 41, 65, the codes in Comalada’s paper [1] are also given.) A
curve with 0 such as 7A20 is its conjugate without 0 such as 7A2. Note that,
for example,
26A10 is not on the tables, since 26A1 and 26A10 are isomorphic
√
over Q( 26). For the same reason, 37A10 , 65B10 , 65B20 and 65A20 are not on
the tables.
(2) The coefficients a1 , a2 , a3 , a4 and a6 are the ones of a defining equation
of each curve.
(3) The discriminant of E is denoted by ∆. In all cases, we denote by ε
the fundamental unit greater than 1. Note that 65A4, 65A40 , 65B1 and 65B2
have no global minimal models and so ∆ cannot be units.
(4) The j-invariant of E is denoted by j.
Table 1: Z/4Z-admissible
463
Elliptic curves
Code
7A2 (E9 )
7A20 (E10 )
7A3 (E14 )
7A30 (E13 )
41A2 (E26 )
41A20 (E25 )
65A4 (E35 )
65A40 (E36 )
a1
1
1 √
−8 + 3√7
−8 − 3 7
1
1
1
1
a2
√
32 + 12√7
32 − 12 √7
−16 + 6√7
−16 √
−6 7
(7 − √41)/2
(7 + √41)/2
−(1 + √65)/2
−(1 − 65)/2
a3
0
0
0
0
√
(7 − √41)/2
(7 + √41)/2
(1 + √65)/2
(1 − 65)/2
a4
√
−10112 − 3822√7
−10112 + 3822
7
√
127 − 48√7
127 +√
48 7
6 − √41
6+ √
41
38 + 4√65
38 − 4 65
a6
√
385060 + 145539√7
385060 − 145539 7
0
0
0
0 √
−(431 + 53√65)/2
−(431 − 53 65)/2
∆
ε3
ε03
−ε06
−ε6
−ε0
−ε
−56 ε3
−56 ε03
Table 2: Z/5Z-admissible
Code
26A1
37A1
a1
√
− 26
0
a2
√
−1 +
√ 26
(19 + 37)/2
a3
√
1 + √26
−6 − 37
a4
√
10 + 4√ 26
(67 + 11 37)/2
a6
√
6 + 2 26
0
∆
−ε6
ε6
j
26
12
2
Table 3: Z/6Z-admissible
Code
6A1 (E1 )
6A10 (E2 )
6A3 (E5 )
6A30 (E6 )
a1
√
√6
−√6
−√6
− 6
a2
√
−2 − √6
−2 +√ 6
7 − √6
7+ 6
a3
−1
−1
√
7 − √6
7+ 6
a4
0
0√
3 − √6
3+ 6
a6
0
0 √
−7 + 3√6
−7 − 3 6
∆
ε3
ε03
ε05
ε5
j
203
203
64(4ε4 + 1)3 ε04
64(4ε04 + 1)3 ε4
Table 4: Z/2Z × Z/2Z-admissible
Code
7A1 (E7 )
0
7A10 (E7
)
41A1 (E23 )
0
41A1 (E24 )
65A1 (E29 )
65B1 (E30 )
65B2 (E32 )
a1
1
1
1
1
1
1
1
a2
√
32 + 4√7
32 −
4
7
√
(7 − √41)/2
(7 + 41)/2
√
−32 − 4 √
65
−159 − 20
√ 65
81 + 10 65
a3
1
1√
7 − √41
7 + 41
0
0
0
a4
√
8 + 3√7
8 − 3√ 7
6 − 4√41
6 + 4 √41
−8 − 65
√
−200 − 25 √65
3225 + 400 65
a6
0
0
0
0
0
0
0
∆
ε6
ε06
−ε6
−ε06
ε04
(5ε)6
(5ε)6
j
2573
2573
−153
−153
−(ε − 16)3 ε0
2573
173
Table 5: Z/2Z × Z/4Z-admissible
Code
65A2 (E31 )
3
a1
1
a2
√
16 + 65
a3
0
a4
√
129 + 16 65
a6
0
∆
ε6
j
173
Proof of Theorem
The following lemma is crucial to prove Theorem.
Lemma 3.1. If an elliptic curve over a number field K has everywhere good
reduction over K, then its j-invariant is an integer of K.
Proof. Let E be an elliptic curve defined over K and let j be the j-invariant of
E. Suppose that E has good reduction at a prime ideal p of K. Then we can
choose a model of E with p-integer coefficients and thus p-unit discriminant
∆. Hence j = c34 /∆ (here c4 is the usual invariant) is a p-integer of K.
Therefore all G-admissible curves appear in the tables in [11] already when
∼
G 6= {0}, Z/2Z and Z/3Z. Hence, to prove Theorem, it is enough to select all
G-admissible curves from the tables. But they are too large to check throughly.
To reduce the amount of consideration, we use following two lemmas:
Lemma 3.2. Let j be the j-invariant of an elliptic curve with everywhere
good reduction over a quadratic field K.
(1) The principal ideal (j) is the cube of some ideal.
(2) If j ∈ Z, then j is the cube of some rational integer.
j
(256ε2 + ε0 )3
02
(256ε + ε)3
−153
−153
173 ε
173 ε0
(8 + ε0 )3
(8 + ε)3
464
Takaaki Kagawa
Proof. (1) is essentially proved in the proof of Lemma 3.1.
(2) [15], Theorem 1.
Lemma 3.3. Let K be an imaginary quadratic field. If the class number
of K is prime to 6, then there is no elliptic curve defined over K having
everywhere good reduction.
Proof. [2], Corollary 3.3, [14], Theorem 5 and [16], Theorem 1.9.
Over real quadratic fields, many nonexistence results and determination
results exist. See Ishii [2], Kagawa [3], [5], [6], Kida [8], [9], Kida–Kagawa [10],
Pinch [12], Takeshi [17] and Yokoyama–Shimasaki [18]. Using their results
together with Lemmas 3.2 and 3.3, we can ignore many quadratic fields.
[I] We show only eight Z/4Z-admissible curves exist. There are many
curves in Table 2 in [11] whose torsion subgroups contain Z/4Z. Most curves
cannot have everywhere
good reduction by Lemma 3.2. For over many fields,
√
for example Q( 17), it is shown that there are no curves with everywhere
√
good reduction by [10], Corollary 1 and [5], Theorem 2. And over Q( 33),
in [4], [6], all elliptic curves√with everywhere good reduction are determined
and none of them has a Q( 33)-rational point of order 4. For the remaining
curves, we compute the conductors and check whether they are trivial or not
using the software Sage [13].
[II] Next, we determine Z/5Z-admissible curves. In Table 8 of [11], the
elliptic curves which have integral j-invariant and defined over quadratic fields
with torsion
subgroup
√ contain Z/5Z are classified. We consider the curves
√
and Q( 37) √
only, because there are no elliptic curve over K =
over
√ Q( 26) √
K by
Q( −1), Q( 10) and Q( 13) with everywhere good reduction over √
Lemma 3.3, Example of [2] and Theorem 2 of [5]. Moreover, over K = Q( 65),
there are no Z/5Z-admissible curves. (See [4].) Finally, other than 26A1 and
37A1, they have shown not to have everywhere good reduction using Sage.
The remaining curves 26A1 and 37A1 have everywhere
good reduction.
Note
√
√
that 26A1 (resp. to 37A1) is isomorphic over Q( 26) (resp. Q( 37)) to 26A10
(resp. 37A10 ).
[III] The case of Z/6Z-admissible curves determined in a similar way.
∼
[IV] In [11], there are eight curves
√ with E(K)
√ tors = Z/8Z and integral jinvariant. They are defined over Q( 5) and Q( 17). But there are no elliptic
curves wnith everywhere good reduction over these fields. (See [2], [5], [10]
and [12].) Thus there are no Z/8Z-admissible curves.
Elliptic curves
465
[V] Nonexistence of Z/7Z- and Z/10Z-admissible curves follows from Lemma
3.3.
[VI] There are no Z/3Z×Z/3Z-admissible curves because in Table 7 in [11],
the elliptic curves with integral j-invariants
√ defined over with torsion group
isomorphic Z/3Z × Z/3Z are defined Q( −3). But by Lemma 3.3, all these
curves do not have everywhere good reduction.
[VII] Finally, we determine Z/2Z × Z/2Z- and Z/2Z × Z/4Z-admissible
curves and show nonexistence of Z/2Z × Z/6Z-admissible curves.
When an elliptic curve E over a quadratic field K having four 2-division
K-rational points and integral j-invariant, the quadratic twists of E have the
same property as E. Moreover, Table 1 in [11], where such curves listed modulo
twists, is huge. (9 pages long!) But the following lemma allows us to skip it:
√
Lemma 3.4. Let K = Q( d) be a quadratic field (real or imaginary). Let E
be an elliptic curve defined over K with everywhere good reduction satisfying
Z/2Z × Z/2Z ⊂ E(K)tors . Then we must have d = 7, 41 or 65 and E is
isomorphic to either 7A2, 7A20 , 7A3, 7A30 , 41A2, 41A20 , 65A2 or 65A20 .
Proof. [1], Theorem 2.
Computing the torsion subgroups by Sage, we can know that only 65A2
has a torsion point of order 4, that is Z/2Z × Z/4Z-admissible and the others
Z/2Z×Z/2Z-admissible. Z/2Z×Z/6Z-admissible curves do not exist, because
in Table 6 of [11], there are four curves with
isomorphic to
√
√ torsion subgroups
Z/2Z × Z/6Z. They are defined over Q( −3) and Q( 3). But over these
fields, there are no elliptic curve with everywhere good reduction. (Lemma
3.3, Theorem 2 of [5] and Theorem 2 of [8].) Therefore there are no elliptic
Z/2Z × Z/6Z-admissible curves over quadratic fields.
Theorem is now proved.
Acknowledgements. The author would like to thank Nao Takeshi and
Shun’ichi Yokoyama for the sevearal useful discussions. In particular he thanks
about the information about the curve 26A1.
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Received: September 15, 2016; Published: October 15, 2016