Math. J. Okayama Univ. 53 (2011), 75–82

TORSION OF ELLIPTIC CURVES OVER QUADRATICCYCLOTOMIC FIELDS
Filip Najman
Abstract. In this√ paper we study the possible torsions of elliptic curves
over Q(i) and Q( −3).

1. Introduction
For an elliptic curve E over a number field K, it is well known, by theMordell-Weil theorem, that the set E(K) of K-rational points on E is a
finitely generated abelian group. The group E(K) is isomorphic to T ⊕ Zr ,where r is a non-negative integer and T is the torsion subgroup. When
K = Q, by Mazur’s Theorem, the torsion subgroup is either cyclic of orderm, where 1 ≤ m ≤ 10 or m = 12, or of the form Z2 ⊕ Z2m , where 1 ≤ m ≤ 4.
If K is a quadratic field, then the following theorem classifies the possible
torsions.
Theorem(Kamienny, [5], Kenku and Momose, [6]). Let K be a quadratic
field and E an elliptic curve over K. Then the torsion subgroup E(K)torsof E(K) is isomorphic to one of the following 26 groups:Zm , for 1 ≤ m ≤ 18, m 6= 17,Z2 ⊕ Z2m , for 1 ≤ m ≤ 6,Z3 ⊕ Z3m , for m = 1, 2Z4 ⊕ Z4 .

Moreover, the only quadratic field over which torsion Z4 ⊕ Z4 appears isQ(i) and the√only quadratic field over which torsions Z3 ⊕ Z3 and Z3 ⊕ Z6
appear is Q( −3).
In [4], Theorem 3.5, it is proved that if we let the quadratic fields vary,then all of the 26 torsion subgroups appear infinitely often.
In this paper we will take a different approach. We will fix the quadratic
field and then study√ the possible torsions. The fields that we are going tostudy, Q(i) and Q( −3), are somewhat special, as they are the only fields
containing roots of unity apart from 1 and −1, i.e. they are the only cyclotomic quadratic fields. Also, over each of these fields, torsion subgroups
appear that appear over no other fields. Note that the rings of integers ofboth these fields are unique factorization domains.
Mathematics Subject Classification. Primary 11G05; Secondary 14G05.
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The results obtained for elliptic curves over Q(i) are presented in the following theorem.
Theorem 1.
(i) Let E be an elliptic curve with rational coefficients.
Then E(Q(i))tors is either one of the groups from Mazur’s Theorem or Z4 ⊕ Z4 .
(ii) Let E be an elliptic curve defined over Q(i). Then E(Q(i))tors iseither one of the groups from Mazur’s Theorem, Z4 ⊕ Z4 or Z13 .
The first part of this theorem is the best possible, while for the second
part we believe that Z13 does not appear as a torsion subgroup, but we wereunable to prove this.
Note that the torsion subgroup Z4 ⊕ Z4 appears infinitely often. Ellipticcurves with this torsion can be written in the form
y 2 = x(x + m2 )(x + n2 ), m, n ∈ Z[i],where m2 − n2 is a square in Z[i]. This is an easy corollary of the 2-descentproposition (see [7], Theorem 4.2, p. 85).
√The results obtained for elliptic curves over Q( −3) are presented in thefollowing theorem.
Theorem 2.
(i)√Let E be an elliptic curve with rational coefficients.
Then E(Q( −3))tors is either one of the groups from Mazur’s Theorem, Z3 ⊕ Z3 or Z3 ⊕ Z6 .
√(ii) Let E be an√elliptic curve defined over Q( −3).
Then E(Q( −3))tors is either one of the groups from Mazur’s Theorem, Z3 ⊕ Z3 , Z3 ⊕ Z6 , Z13 or Z18 .
Again, the first part of this theorem is the best possible(Z3 ⊕ Z3 andZ3 ⊕ Z6 appear infinitely often), while for the second part we believe that
Z13 and Z18 do not appear as torsion subgroups, but we were unable toprove this.

2. Torsion over Q(i)Throughout this chapter, the following extension of the Lutz-Nagell Theorem is used to compute torsion groups of elliptic curves.
Theorem (Extended Lutz-Nagell Theorem). Let E : y 2 = x3 + Ax + B withA, B ∈ Z[i]. If a point (x, y) ∈ E(Q(i)) has finite order, then
(1) x, y ∈ Z[i].
(2) Either y = 0 or y 2 |4A3 + 27B 2 .

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The proof of the Lutz-Nagell Theorem can easily be extended to ellipticcurves over Q(i). Details of the proof can be found in [12], Chapter 3. An
implementation in Maple can be found in [12], Appendix A.
Next, we give a result that applies to elliptic curves over all quadratic fields
and that is an immediate corollary of the main result of [8] (see also [2]).
Lemma 3. Let E be an elliptic√ curve with rational coefficients and d asquare-free integer. Then E(Q( d))tors cannot be Z11 , Z13 or Z14 .
Next we give a series of lemmas that will prove Theorem 1.
Lemma 4. E(Q(i))tors cannot be Z11 , Z18 or Z2 ⊕ Z10 .
Proof: It is proved in [6], Example 3.2 that the torsion of an elliptic curve
over Q(i) cannot be Z2 ⊕Z10 . It is proved in [10], Theorem 2 that the torsion
cannot be Z11 . Since the rational prime 3 remains prime in Z[i], conditioni) of Proposition 2.4 from [6] is satisfied and hence, the torsion cannot beZ18 .
Lemma 5. E(Q(i))tors cannot be Z16 .
Proof : By [9], case 2.5.5., page 37, we see that elliptic curves with torsion
Z16 over Q(i) are induced by the solutions of the equation(1)where t satisfies(2)

s2 = t(t2 + 1)(t2 + 2t − 1), s, t ∈ Q(i),t(t4 − 1)(t2 + 2t − 1)(t2 − 2t − 1) 6= 0.

It follows that it is enough to prove that this equation has no solutions. Let(3)t = α,(4)

t2 + 1 = βand(5)

t2 + 2t − 1 = γ,where α, β and γ are square-free, nonzero Gaussian integers. Also, let t = uv ,where u and v are coprime, nonzero Gaussian integers. First, we prove that
α is relatively prime to β and γ. Suppose a Gaussian prime π divides bothα and β. From (3) it follows that π divides either u or v an odd numberof times. From (4), it follows that π divides u2 + v 2 , and since it divides
exactly one of u, v, this is impossible. Suppose a Gaussian prime π divides
both α and γ. Again, π divides exactly one of u, v. From (5), it follows that
π divides u2 + 2uv − v 2 , which is again impossible. Since −1 is a square inZ[i], we conclude that α ∈ {1, i}.

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Next, we prove that gcd(β, γ) = 1 or 1 + i. Let π be a Gaussian prime
dividing both β and γ. By subtracting (4) from (5), we conclude that πdivides 2uv − 2v 2 = 2v(u − v). As it was already proved, since π divides
β, π cannot divide u, implying π|2(u − v). Suppose π|(u − v), i.e. u ≡ v(mod π). Now, (4) implies 2u2 ≡ 0 (mod π), again implying π|2. Since
2 = −i(1 + i)2 , we conclude that the only possibilities for β and γ areβ, γ ∈ {1, i, 1 + i, i(1 + i)}.
We assert that none of these are possible. Suppose β = 1. Since α = 1 or222 = ± x4 . Multiplying (4) by y 4 wei, we can write t = xy2 or t = ix
,i.e.
t
2yy4
442get x ± y = ±z . It was proved by Hilbert(see [3]) that this equation has
only trivial solutions in Gaussian integers, implying t = 0.
Suppose β = i. Multiplying (3) and (4) we obtain iy 2 = t3 + t or −y 2 =t3 +t, leading to elliptic curves in Weierstrass form y 2 = x3 −x and y 2 = x3 +xrespectively. Using the program [11], written in PARI, we compute that therank of this curve is 0. It is easy to compute, using the Extended LutzNagell Theorem that the torsion subgroup of both these curves over Q(i) isZ2 ⊕ Z2 . All the torsion points of these curves satisfy t(t4 − 1) = 0.
Suppose β = 1 + i or i(1 + i). Multiplying (3) and (4) we obtain oneof the following three elliptic curves (1 + i)y 2 = t3 + t, i(1 + i)y 2 = t3 + tand −(1 + i)y 2 = t3 + t. These curves induce the curves y 2 = x3 + 2ix andy 2 = x3 − 2ix in Weierstrass form, both of them having rank 0 (again, thisis computed using [11]) and torsion Z2 ⊕ Z2 . All of the torsion points inducet satisfying t(t4 − 1) = 0. Hence, the starting equation has no solutions. Lemma 6. E(Q(i))tors cannot be Z15 .
Proof : By [9], case 2.5.4, pages 34 and 35, elliptic curves with torsion
subgroups isomorphic to Z15 are induced by the solutions over Q(i) of(6)s2 + st + s = t3 + t2

satisfyingt(t + 1)(t2 + t + 1)(t4 + 3t3 + 4t2 + 2t + 1)(t4 − 7t3 − 6t2 + 2t + 1) 6= 0.
Using [11], we compute that the rank of (6) over Q(i) is 0 and the torsion
points give t = 0 or −1, implying that the equation has no solutions.
Lemma 7. E(Q(i))tors cannot be Z2 ⊕ Z12 .
Proof: By [9], case 2.5.8, pages 42–44, elliptic curves with torsion Z2 ⊕Z12
are induced by the solutions over Q(i) of the equation(7)

s2 = (2t2 − 2t + 1)(6t2 − 6t + 1)

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79

satisfying(8)t(t − 1)(2t − 1)(2t2 − 2t + 1)(3t2 − 3t − 1)(6t2 − 6t + 1) 6= 0.

The elliptic curve (7) has the Weirstrass form

y 2 = x3 − x2 + x.

This curve has rank 0 and 8 torsion points. They are {O, (0, 0), (1, ±1),(±i, ±1)} in Weierstrass form, inducing t = 0, 1, 12 or 1±i
2 , none of themsatisfying (8).
Lemma 8. E(Q(i))tors cannot be Z14 .
Proof: By [9], case 2.5.3, page 31, elliptic curves with torsion Z14 areinduced by the solutions over Q(i) of the equation
satisfying

s2 + st + s = t3 − tt(t2 − 1)(t3 − 9t2 − t + 1)(t3 − 2t2 − t + 1) 6= 0.
The given curve has rank 0 and 6 torsion points over Q(i), all of themsatisfying t = 0 or ±1.
Lemmas 4, 5, 6, 7 and 8 prove Theorem 1, (ii). Combining this withLemma 3, we get Theorem 1, (i).
√3. Torsion over Q( −3)As some of the proofs in this section are similar to the ones
√ in the previous
1− −3
section, some technical details will be omitted. Let ω =. It is easy to2
see thatωisaprimitive
sixth
rootofunityandZ[ω]istheringof integers√of Q( −3).
√Lemma 9. E(Q( −3))tors cannot be Z14 , Z15 , or Z2 ⊕ Z12 .
Proof: It is proved in [6] that the torsion subgroup cannot be Z14 (Example2.5) or Z2 ⊕ Z12 (Example 3.2). The proof that the torsion cannot be Z15 iscompletely analogous to the proof of Lemma 7.
√Lemma 10. E(Q( −3))tors cannot be Z11 .
Proof:
As can be seen in [9], case 2.5.1, page 25, the solutions s, t ∈√Q( −3) of the equations2 − s = t3 − t2satisfyingt(t − 1)(t5 − 18t4 + 35t3 − 16t2 − 2t + 1) 6= 0

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FILIP NAJMAN√
induce elliptic curves with torsion Z11 over Q( −3). The rank of this curveis 0 and there are 5 torsion points, all of them satisfying t = 0 or 1 (see [9],Lemma 2.1).
√Lemma 11. E(Q( −3))tors cannot be Z2 ⊕ Z10 .
Proof:
As can be seen in [9], case 2.5.7, pages 39–40, the solutions s, t ∈√Q( −3) of the equation
s2 = t3 + t2 − tsatisfyingt(t2 − 1)(t2 − 4t − 1)(t2 + t − 1) 6= 0√
induce elliptic curves with torsion Z2 ⊕ Z10 over Q( −3). The rank of thiscurve is 0 and there are 6 torsion points, all of them satisfying t = 0, −1 or1 (see [9], Lemma 2.4).
As in the proof of Theorem 1, the hardest part of the proof of Theorem
2 is eliminating the possibility of the torsion being Z16 .
√Lemma 12. E(Q( −3))tors cannot be Z16 .
Proof: Again, we have to prove that the equation (1) has no solutions
satisfying (2). We follow the same strategy of the proof of Lemma 6, anddefine α, β and γ in the same way. It can be proved in the same way asin Lemma 6 that α is a unit and that each of β and γ is either a unit
or twice a unit. √As every unit is a square or ω times a square, and as
γ = αβ (mod (Q( −3)∗ )2 ), we see that we have 8 possibilities for the triples(α, β, γ). If t is a solution of (1), then t has to be the first coordinate of apoint on both curves
E1 : αβy 2 = t3 + t
andE2 : αγy 2 = t3 + 2t2 − t.

In the following table we give the ranks of these curves depending on α andβ.
√√α βγ rank(E1 (Q( −3)) rank(E2 (Q( −3))
1 11101 ω ω10
1 22021 2ω 2ω00
ω 1 ω10ω ω110
ω 2ω 200
ω 2 2ω02

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As it can be seen, for each case there either E1 or E2 has rank 0, and thus
only the torsion points are possible solutions. It remains to check the torsion
points of the following four curves:(9)y 2 = x3 + 2x2 − x,(10)y 2 = x3 + 2ωx2 − ω 2 x,

(11)y 2 = x3 + 4x,(12)y 2 = x3 + 4ω 2 x.

The torsion groups were now computed in APECS([1]). The torsion of thecurves (11) and (12) is Z4 , corresponding to t = 0, ±1 on the curve E1 . The
torsion of the curves (9) and (10) is Z2 , corresponing to t = 0 on the curve
E2 . We obtain that none of the t induced by the torsion points satisfies (2).
We conclude that the torsion Z16 is impossible.
Lemmas 9, 10, 11 and 12 combined with Lemma 3 prove Theorem 2.
Remark. In order√to prove that there are no elliptic curves with torsion Z13
over Q(i) and Q( −3), one would have to prove that there are no solutionsin the respective quadratic field of the equation(13)

s2 = t6 − 2t5 + t4 − 2t3 + 6t2 − 4t + 1

satisfyingt(t − 1)(t3 − 4t2 + t + 1) 6= 0.

Similarly,to prove that there are no elliptic curves with torsion Z18√over√Q( −3) one would have to prove that there do not exist s, t ∈ Q( −3)satisfying(14)

s2 = t6 + 2t5 + 5t4 + 10t3 + 10t2 + 4t + 1

andt(t + 1)(t2 + t + 1)(t3 − 3t − 1) 6= 0.

Note that (13) and (14) are both hyperelliptic curves of genus 2.
Acknowledgements
The author would like to thank Andrej Dujella for motivating and helpful
discussions. Also, the author is grateful to Mirela Jukić-Bokun and thereferee for pointing out some mistakes in earlier versions of this manuscript.

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Department of Mathematics
University of Zagreb
Bijenička cesta 30, 10000 Zagreb
Croatiae-mail address: fnajman@math.hr(Received June 5, 2009 )(Revised August 2, 2009 )