In this paper we study the possible torsions of elliptic curves over $\mathbb
Q(i)$ and $\mathbb Q(\sqrt {-3})$.Comment: 9 pages, to appear in Math. J. Okayama Univ

Najman, Filip

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In this paper we study the possible torsions of elliptic curves over ℚ(i) and ℚ(√−3)

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TORSION OF ELLIPTIC CURVES OVER QUADRATIC CYCLOTOMIC FIELDS

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Math. J. Okayama Univ. 53 (2011), 75–82TORSION OF ELLIPTIC CURVES OVER QUADRATICCYCLOTOMIC FIELDSFilip NajmanAbstract. In this paper we study the possible torsions of elliptic curvesover Q(i) and Q(√−3).1. IntroductionFor 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 afinitely 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. WhenK = 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 possibletorsions.Theorem(Kamienny, [5], Kenku and Momose, [6]). Let K be a quadraticfield 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 ⊕ Z6appear 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 quadraticfield 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 fieldscontaining roots of unity apart from 1 and −1, i.e. they are the only cy-clotomic quadratic fields. Also, over each of these fields, torsion subgroupsappear 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 14G05.7576 FILIP NAJMANThe results obtained for elliptic curves over Q(i) are presented in the fol-lowing 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 Theo-rem 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 secondpart 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 formy2 = 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 The-orem, 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 The-orem, 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 thatZ13 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 The-orem is used to compute torsion groups of elliptic curves.Theorem (Extended Lutz-Nagell Theorem). Let E : y2 = 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 y2|4A3 + 27B2.TORSION OF ELLIPTIC CURVES OVER QUADRATIC CYCLOTOMIC FIELDS 77The 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. Animplementation in Maple can be found in [12], Appendix A.Next, we give a result that applies to elliptic curves over all quadratic fieldsand 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 curveover Q(i) cannot be Z2⊕Z10. It is proved in [10], Theorem 2 that the torsioncannot 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 torsionZ16 over Q(i) are induced by the solutions of the equation(1) s2 = t(t2 + 1)(t2 + 2t− 1), s, t ∈ Q(i),where t satisfies(2) 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 = βand(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 thatα is relatively prime to β and γ. Suppose a Gaussian prime pi divides bothα and β. From (3) it follows that pi divides either u or v an odd numberof times. From (4), it follows that pi divides u2 + v2, and since it dividesexactly one of u, v, this is impossible. Suppose a Gaussian prime pi dividesboth α and γ. Again, pi divides exactly one of u, v. From (5), it follows thatpi divides u2 + 2uv − v2, which is again impossible. Since −1 is a square inZ[i], we conclude that α ∈ {1, i}.78 FILIP NAJMANNext, we prove that gcd(β, γ) = 1 or 1 + i. Let pi be a Gaussian primedividing both β and γ. By subtracting (4) from (5), we conclude that pidivides 2uv − 2v2 = 2v(u − v). As it was already proved, since pi dividesβ, pi cannot divide u, implying pi|2(u − v). Suppose pi|(u − v), i.e. u ≡ v(mod pi). Now, (4) implies 2u2 ≡ 0 (mod pi), again implying pi|2. Since2 = −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 ori, we can write t = x2y2or t = ix2y2, i.e. t2 = ±x4y4. Multiplying (4) by y4 weget x4 ± y4 = ±z2. It was proved by Hilbert(see [3]) that this equation hasonly trivial solutions in Gaussian integers, implying t = 0.Suppose β = i. Multiplying (3) and (4) we obtain iy2 = t3 + t or −y2 =t3+t, leading to elliptic curves inWeierstrass form y2 = x3−x and y2 = 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 Lutz-Nagell 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)y2 = t3 + t, i(1 + i)y2 = t3 + tand −(1 + i)y2 = t3 + t. These curves induce the curves y2 = x3 + 2ix andy2 = 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 torsionsubgroups isomorphic to Z15 are induced by the solutions over Q(i) of(6) s2 + st+ s = t3 + t2satisfyingt(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 torsionpoints 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⊕Z12are induced by the solutions over Q(i) of the equation(7) s2 = (2t2 − 2t+ 1)(6t2 − 6t+ 1)TORSION OF ELLIPTIC CURVES OVER QUADRATIC CYCLOTOMIC FIELDS 79satisfying(8) t(t− 1)(2t− 1)(2t2 − 2t+ 1)(3t2 − 3t− 1)(6t2 − 6t+ 1) 6= 0.The elliptic curve (7) has the Weirstrass formy2 = 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, 12or 1±i2, 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 equations2 + st+ s = t3 − tsatisfyingt(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 previoussection, some technical details will be omitted. Let ω = 1−√−32. It is easy tosee that ω is a primitive sixth root of unity and Z[ω] is the ring of integersof 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= 080 FILIP NAJMANinduce 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 equations2 = t3 + t2 − tsatisfyingt(t2 − 1)(t2 − 4t− 1)(t2 + t− 1) 6= 0induce 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 Theorem2 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 solutionssatisfying (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 unitor twice a unit. As every unit is a square or ω times a square, and asγ = 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 curvesE1 : αβy2 = t3 + tandE2 : αγy2 = t3 + 2t2 − t.In the following table we give the ranks of these curves depending on α andβ.α β γ rank(E1(Q(√−3)) rank(E2(Q(√−3))1 1 1 1 01 ω ω 1 01 2 2 0 21 2ω 2ω 0 0ω 1 ω 1 0ω ω 1 1 0ω 2ω 2 0 0ω 2 2ω 0 2TORSION OF ELLIPTIC CURVES OVER QUADRATIC CYCLOTOMIC FIELDS 81As it can be seen, for each case there either E1 or E2 has rank 0, and thusonly the torsion points are possible solutions. It remains to check the torsionpoints of the following four curves:(9) y2 = x3 + 2x2 − x,(10) y2 = x3 + 2ωx2 − ω2x,(11) y2 = x3 + 4x,(12) y2 = x3 + 4ω2x.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. Thetorsion of the curves (9) and (10) is Z2, corresponing to t = 0 on the curveE2. 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 Z13over 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+ 1satisfyingt(t− 1)(t3 − 4t2 + t+ 1) 6= 0.Similarly, to prove that there are no elliptic curves with torsion Z18 overQ(√−3) one would have to prove that there do not exist s, t ∈ Q(√−3)satisfying(14) s2 = t6 + 2t5 + 5t4 + 10t3 + 10t2 + 4t+ 1andt(t+ 1)(t2 + t+ 1)(t3 − 3t− 1) 6= 0.Note that (13) and (14) are both hyperelliptic curves of genus 2.AcknowledgementsThe author would like to thank Andrej Dujella for motivating and helpfuldiscussions. Also, the author is grateful to Mirela Jukic´-Bokun and thereferee for pointing out some mistakes in earlier versions of this manuscript.82 FILIP NAJMANReferences[1] I. Connell, APECS, ftp://ftp.math.mcgill.ca/pub/apecs/.[2] Y. Fujita, Torsion subgroups of elliptic curves in elementary abelian 2-extensions ofQ, J. Number Theory 114 (2005), 124–134.[3] D. Hilbert, Jahresbericht d. Deutschen Math.-Vereinigung, 4, 1894/1895, 517–525.[4] D. Jeon, C. H. Kim and E. Park, On the torsion of elliptic curves over quartic numberfields, J. London Math. Soc. (2) 74 (2006), 1–12.[5] S. Kamienny, Torsion points on elliptic curves and q-coefficients of modular forms,Invent. Math. 109 (1992), 221–229.[6] M. A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadraticfields, Nagoya Math. J. 109 (1988), 125–149.[7] A. Knapp, Elliptic curves, Princeton Univ. Press, 1992.[8] M. Laska and M. Lorenz, Rational points on elliptic curves over Q in elementaryabelian 2-extensions of Q, J. Reine Angew. Math. 355 (1985), 163–172.[9] F. P. Rabarison, Torsion et rang des courbes elliptiques definies sur les corps de nom-bres algebriques, Doctorat de Universite de Caen, 2008.[10] M. A. Reichert, Explicit Determination of Nontrivial Torsion Structures of EllipticCurves Over Quadratic Number Fields, Math. Comp. 174 (1986), 637–658.[11] D. Simon, Le fichier gp, http://www.math.unicaen.fr/∼simon/ell.gp.[12] T. Thongjunthug, Elliptic curves over Q(i), Honours thesis (2006).Department of MathematicsUniversity of ZagrebBijenicˇka cesta 30, 10000 ZagrebCroatiae-mail address: fnajman@math.hr(Received June 5, 2009 )(Revised August 2, 2009 )

Elliptic curves over Q(i), Honours thesis

Elliptic curves,

Explicit Determination of Nontrivial Torsion Structures of Elliptic Curves Over Quadratic Number Fields,

Jahresbericht d.

Le ﬁchier gp,

On the torsion of elliptic curves over quartic number ﬁelds,

Rational points on elliptic curves over Q in elementary abelian 2-extensions of

Torsion et rang des courbes elliptiques deﬁnies sur les corps de nombres algebriques, Doctorat de Universite de Caen,

Torsion points on elliptic curves and q-coeﬃcients of modular forms,

Torsion points on elliptic curves deﬁned over quadratic ﬁelds,

Torsion subgroups of elliptic curves in elementary abelian 2-extensions of

Filip Najman

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Torsion of elliptic curves over quadratic cyclotomic fields

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Torsion of elliptic curves over quadratic cyclotomic fields

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