Torsion of rational elliptic curves over quadratic fields

arXiv:1411.3471v1 [math.NT] 13 Nov 2014 TORSION OF RATIONAL ELLIPTIC CURVES OVER QUADRATIC FIELDS ENRIQUE GONZÁLEZ–JIMÉNEZ AND JOSÉ M. TORNERO Abstract. Let Ebe an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)tors and the torsion subgroup E(K)tors, where Kis a quadratic number field. 1. Introduction Let Ebe an elliptic curve defined over a number field K. The Mordell-Weil Theorem states that the set of K-rational points, E(K), is a finitely generated abelian group. So it can be written as E(K)≃E(K)tors ⊕Zr, for some nonnegative integer r(rank of E(K)) and some finite torsion subgroup E(K)tors. It is well known that there exist two positive integers n, m such that E(K)tors is isomorphic to Cn×Cm, where Cnis the cyclic group of order n[24]. This paper focuses on a particular problem concerning the torsion part, which we will explain now. We define sets S(d)and Φ(d)as follows: •S(d)is the set of primes that can appear as the order of a torsion point of an elliptic curve Edefined over a number field of degree d. •Φ(d)is the set of possible groups that can appear as the torsion subgroup of an elliptic curve defined over a certain number field Kof degree d. Mazur’s landmark papers [17, 18] established that S(1) = {2,3,5,7}and Φ(1) = {Cn|n= 1,...,10,12}∪{C2×C2m|m= 1,...,4}. After this, a long series of papers by Kenku, Momose and Kamienny ending in [10, 12] slowly unfolded the quadratic case to finally reach a full description of S(2) = {2,3,5,7,11,13}and Φ(2): Φ(2) = {Cn|n= 1,...,16,18}∪{C2×C2m|m= 1,...,6}∪ {C3×C3r|r= 1,2}∪{C4×C4}. We do not have, as of today, such a precise description of Φ(d)for d≥3although work by Parent [22] has obtained S(3) and Derickx, Kamienny, Stein and Stoll have announced [4] that they have established the sets S(d)for d= 4,5. A fundamental result here is the celebrated Uniform Boundness Theorem, a long–standing conjecture finally proved by Merel [19], which states that there exists a constant B(d) such that |G| ≤ B(d), for all G∈Φ(d). Although Merel’s proof was not explicit, further versions (an Oesterlé’s 1994 unpublished paper and Parent [23]) have given precise values for B(d). 2010 Mathematics Subject Classification. Primary: 11G05,11R11; Secondary:14G05. Key words and phrases. Elliptic curves, Torsion subgroup, rationals, quadratic fields. The first author was partially supported by the grant MTM2012–35849. The second author was partially supported by the grants FQM–218 and P08–FQM–03894, FSE and FEDER (EU). 1 2 ENRIQUE GONZÁLEZ–JIMÉNEZ AND JOSÉ M. TORNERO Let us fix some useful notations: •Let Ebe an elliptic curve defined over a number field. Without loss of generality we can assume Eis defined by a short Weierstrass form E:Y2=X3+AX +B;A, B ∈K, and we will write, E(K) = (x, y)∈K2|y2=x3+Ax +B∪{O}, the set of K–rational points of E, and Oits point at infinity. •Let SQ(d)be the set of primes that can appear as the order of a torsion point defined over a number field of degree d, on an elliptic curve Edefined over the rationals. •Let ΦQ(d)be the set of possible groups that can appear as the torsion subgroup over a number field of degree d, of an elliptic curve Edefined over the rationals. •For an elliptic curve E, let ∆Ebe, as customary, its discriminant. •For an elliptic curve Eand an integer n, let E[n]be the subgroup of all points whose order is a divisor of n(over Q), and let E(K)[n]be the set of points in E[n]with coordinates in K, for any number field K(including the case K=Q). •Under the same conditions, let Q(E[n]) be the extension generated by all the coordinates of points in E[n]. •For an elliptic curve Edefined over the rationals given by a short Weierstrass equation E:Y2=X3+AX +B, and a square–free integer D, let EDdenote its quadratic twist. That is, the elliptic curve with a Weierstrass equation ED:DY 2=X3+AX +B. Please mind that, in the sequel, for examples and precise curves we will use the Antwerp–Cremona tables and labels [1, 3]. The set SQ(d)is known for d≤42, and there is a conjectural description by Lozano–Robledo [16] which encompasses both the known cases and experimental data. For example, SQ(1) = SQ(2) = {2,3,5,7}. The sets ΦQ(d)have been completely described by Najman [20] for d= 2,3. Concretely, he proved: ΦQ(2) = {Cn|n= 1,...,10,12,15,16}∪{C2×C2m|m= 1,...,6}∪ {C3×C3r|r= 1,2}∪{C4×C4}, ΦQ(3) = {Cn|n= 1,...,10,12,13,14,18,21}∪{C2×C2m|m= 1 . . . , 4,7}. The remarkable fact is that although the set Φ(3) is still unknown, Najman has determined the set ΦQ(3). It is also worth noting that Fujita [6] has explicitely determined the torsion subgroups over the maximal elementary 2–extension of Q(that is Q({√m|m∈Z}) that may arise from an elliptic curve defined over Q. This classification might be of great help in the open problems that we will pose later. Definition 1. Let G∈Φ(1). We will write ΦQ(d, G)the set of possible groups that can appear as the torsion subgroup over a certain number field Kof degree d, of an elliptic curve Edefined over the rationals, such that E(Q)tors =G. TORSION OF RATIONAL ELLIPTIC CURVES OVER QUADRATIC FIELDS 3 Our aim in this paper is, first to compute and then to understand better, ΦQ(2, G). That is, the behaviour of a particular torsion group of Φ(1) when we enlarge the base field Qby means of a quadratic extension. In order to guess what ΦQ(2, G)may look like, we carried out an exhaustive computation, obtaining the groups E(K)tors, for all curves Edefined over the rationals with conductor less than 300000 from [3]. The main result of this paper is the following: Theorem 2. For G∈Φ(1), the set ΦQ(2, G)is the following: GΦQ(2, G) C1{C1,C3,C5,C7,C9} C2{C2,C4,C6,C8,C10 ,C12 ,C16 ,C2×C2,C2×C6,C2×C10} C3{C3,C15,C3×C3} C4{C4,C8,C12 ,C2×C4,C2×C8,C2×C12 ,C4×C4} C5{C5,C15} C6{C6,C12,C2×C6,C3×C6} C7{C7} C8{C8,C16,C2×C8} C9{C9} C10 {C10,C2×C10} C12 {C12,C2×C12} C2×C2{C2×C2,C2×C4,C2×C6,C2×C8,C2×C12} C2×C4{C2×C4,C2×C8,C4×C4} C2×C6{C2×C6,C2×C12} C2×C8{C2×C8} 2. The set ΦQ(2, G) A very important and partial result of our problem, concretely the case of Gbeing non–cyclic, has already been completely solved by Kwon [14]. More precisely, the result goes as follows: (1) ΦQ(2,C2×C2n) =        {C2×C2m|m= 1,2,3,4,6}if n= 1, {C2×C2m|m= 2,4}∪{C4×C4}if n= 2, {C2×C2m|m= 3,6}if n= 3, {C2×C8}if n= 4. The last three cases were carefully detailed, including necessary and sufficient conditions on the coefficients of a short Weierstrass equation of the elliptic curve and the discriminant of the quadratic field, to determine which of the possible groups actually happened for a given case. The first case, however, lacked of such characterisation and it is determined by ruling the other possibilities out. This appears to be a common fact in the study of torsion subgroups: the simpler the structure is, the more difficult becomes to describe in effective terms (see, for instance, [7]). The next auxiliary result is a particular case of Lemma 1.1 in [15]. However, we need some details from the proof (see [2]). Theorem 3. Let Ebe an elliptic curve defined over Q,Da square-free integer and K=Q(√D). There exists a pair of homomorphisms E(K)Ψ −→ E(Q)×ED(Q)Ψ −→ E(K) 4 ENRIQUE GONZÁLEZ–JIMÉNEZ AND JOSÉ M. TORNERO such that Ψ◦Ψ = [2] and Ψ◦Ψ = [2] ×[2]. Moreover, ker(Ψ) ⊂E(K)[2],ker(Ψ) ⊂E(Q)[2] ×ED(Q)[2] and coker(Ψ), coker(Ψ) are groups where every non–zero element has order 2. Proof. Let σbe the non–trivial element of Gal(K/Q)and α∈K. Let us write, for any P= (x, y), σP = (σ(x), σ(y)) and φ(P, α) = (x, αy). Recall the canonical K–isomorphisms between Eand its D–twist ED: E(K)φ(·,1/√D)ED(K)φ(·,√D)E(K) Let P∈E(K). As σ(P+σP ) = P+σP, we have P+σP ∈E(Q). Consider now P−σP . We have that, either σ(P−σP ) = O,in which case P=σP and P∈E(Q), or σ(P−σP) = −(P−σP )6=O.Should this be the case, if we write R=P−σP = (xR, yR)we have σ(xR, yR) = (xR,−yR). Therefore, yR=zR√D for some zR∈Q, and in this case it is clear that φ(xR, yR),1/√D= (xR, zR)∈ED(Q). So we define Ψas follows: Ψ : E(K)−→ E(Q)×ED(Q) P7−→ P+σP, φ P−σP, 1/√D Ψis clearly a homomorphism: it is straightforward if we write it Ψ : E(K)−→ E(K)×ED(K), and we have just shown Im(Ψ) ⊂E(Q)×ED(Q). On the other hand, let us consider a point R= (xR, yR)∈ED(Q). Clearly φ(R, √D) = xR, yR√D∈E(K), and we define then Ψ : E(Q)×ED(Q)−→ E(K) (P, R)7−→ P+φ(R, √D). Analogously, it is not difficult to see Ψis a homomorphism. We could have defined (exactly the same way) Ψ : E(K)×ED(K)−→ E(K); which is clearly a homomorphism, and then we have it restricted to the subgroup E(Q)×ED(Q). With these definitions, the results from the theorem can be easily deduced.  Corollary 4. Let Ebe an elliptic curve defined over Q,Dan square-free integer and K=Q(√D). If nis odd, then there exists an isomorphism E(K)[n]≃E(Q)[n]×ED(Q)[n]. Our description of ΦQ(2, G), for the eleven cyclic groups in Φ(1), will rest in the following result: Theorem 5. Let Ebe an elliptic curve defined over Q,Ka quadratic number field, G∈ΦQ(1) and H∈ΦQ(2) such that E(Q)tors ≃Gand E(K)tors ≃H. (i) If Q(E[3]) = Q√−3, then C3⊂G. (ii) If C26⊂ G, then C26⊂ H. TORSION OF RATIONAL ELLIPTIC CURVES OVER QUADRATIC FIELDS 5 (iii) If Gis cyclic, C2⊆Gand C46⊂ G, then C2×C46⊂ H. (iv) If G=C4, then H6=C16. (v) If G=C3, then H6=C9. (vi) If H=C15, then G=C3or G=C5. Proof. (i) is Proposition 2.1 from [21]. Note (see Section 3) that if E(K)has full 3–torsion, then K=Q√−3. (ii) Direct, as the hypothesis implies the irreducibility of X3+AX +Bover Q (hence over K). (iii) With no loss of generality, assume Eto have the form E:Y2=X(X2+AX +B). If C2×C2⊂E(K), it must then be K=Q(√∆E), with ∆E=A2−4B, which we will call Dfrom now on. The full set of points with order two is (0,0),−1 2(A−√D),0,−1 2(A+√D),0. Let us assume there is a point of order 4. Then one of the previous points is the double of such a point, hence (see [13] Thm. 4.2.) one of the following pairs consists of two squares in K: 1 2(A−√D),1 2(A+√D),1 2(−A+√D),√D,−√D, −1 2(A+√D). Only the first possibility can hold. Now an element a+b√D∈(K∗)2if and only if there exist x, y ∈Qsuch that a=x2+Dy2, b = 2xy; so we must have A/2 = x2+Dy2 1 = 4xy A/2 = t2+Dz2 −1 = 4tz Clearly both systems have a solution if and only if one of them has. The first gives rise to y=1 4x, x2=A 4±√B 2, so we must assume B=C2and then x2=A/4±C/2. Therefore, our curve must have the form (renaming xand Cas α/2and β, purely for aesthetic purposes): E:Y2=X(X2+ (α2+ 2β)X+β2), but for every Ein this familiy of curves we have C4⊂E(Q), as the point (−β, αβ) has order 4. Note that the non–vanishing of ∆Eis equivalent here to α2(α+4β)6= 0. (iv) Assume we have K=Q(√D)and E, with E(Q)tors ≃ C4,E(K)≃ C16. Then, from Ψ, 0→ker(Ψ) i −→ E(K)Ψ −→ E(Q)×ED(Q)π −→ coker(Ψ) →0, and considering the torsion part, we have ker(Ψ) is either trivial or C2and ED(Q)tors ≃ C2n, with n= 1, . . . , 6. So the only possibility is ker(Ψ) ≃ C2,ED(Q)tors ≃ C8. But then coker(Ψ) ≃ C4, which contradicts Theorem 3. (v) Direct from Corollary 4, with n= 9. (vi) If H=C15, then Najman [20, Theorem 2, c)] has shown that Eis the elliptic curve 50b1 or 50a3 with K=Q(√5); and 50b2 or 450b4 with K=Q(√−15). 6 ENRIQUE GONZÁLEZ–JIMÉNEZ AND JOSÉ M. TORNERO Both 50b1 and 50b2 have G=C5, while 50a3 and 450b4 have G=C3. Therefore, H=C15 will only appear in ΦQ(2, G)for G=C3,C5. Proof. (Theorem 2) The sets ΦQ(2, G)were first conjectured by the computations mentioned above. The relevant results can be found in Table 2. In particular, this shows that all groups said to be in ΦQ(2, G)belong to this set. Table 1. The cases not treated in [14]. The table displays either if the case happens (X), if it is impossible because G6⊂ H(−) or if it is ruled out by Theorem 5 ((i)–(vi)). C1C2C3C4C5C6C7C8C9C10 C12 C1X− − − − − − − − − − C2(ii) X− − − − − − − − − C3X−X− − − − − − − − C4(ii) X−X− − − − − − − C5X− − − X− − − − − − C6(ii) X(ii) − − X− − − − − C7X−−−−−X− − − − C8(ii) X−X− − − X− − − C9X−(v) − − − − − X− − C10 (ii) X− − (ii) − − − − X− C12 (ii) X(ii) X−X− − − − X C15 (vi) −X−X− − − − − − C16 (ii) X−(iv) − − − X− − − C2×C2(ii) X− − − − − − − − − C2×C4(ii) (iii) X− − − − − − − − C2×C6(ii) X(ii) − − X− − − − − C2×C8(ii) (iii) −X− − − X− − − C2×C10 (ii) X− − (ii) − − − − X− C2×C12 (ii) (iii) (ii) X−(iii) − − − − X C3×C3(i) −X− − − − − − − − C3×C6(i), (ii) (i) (ii) − − X− − − − − C4×C4(ii) (iii) −X− − − − − − − The groups Hfrom ΦQ(2) that do not appear in some ΦQ(2, G), with G < H can be ruled out from ΦQ(2, G)most of the times using the previous theorem. Table 1 (row =H, column =G) deals with the case G6=C2× C2m. The non–cyclic case being treated in (1) (cf. [14]).  3. Real vs. imaginary quadratic extensions A natural question might be if there is any substantial difference between the real and imaginary quadratic case. Our computations (see Section 5) show it is not so, except for some well–known cases. In fact, from the Weil pairing [24] we know that, for a number field K, if Cm× Cm⊂E(K), then Kcontains the cyclotomic field generated by the m–th roots of unity. In the quadratic case, that implies: TORSION OF RATIONAL ELLIPTIC CURVES OVER QUADRATIC FIELDS 7 • C3×C3and C3×C6can only appear in the imaginary extension Q√−3. • C4×C4can only appears in the imaginary extension Q√−1. For all the remaining cases, groups in ΦQ(2) appear in both real and imaginary cases. This is shown in Table 2. Consider all elliptic curves, defined over the rationals with: •E(Q)tors as given in the first column •E(Q(√D))tors as given in the second column, for some Das given in the fourth (resp. sixth) column for the real case (resp. imaginary case). Then •The seventh (penultimate) column is the number of curves with conductor less than 300000 which meet this situation, for a real extension Q⊂K. •The eighth (last) column is is the number of curves with conductor less than 300000 which meet this situation, for an imaginary extension Q⊂K. Notice that the only cases where "−" appears are the ones remarked above. 4. On the number of quadratic extensions with proper extension of the torsion subgroup Consider the following problem, closely related to our original one. Take an elliptic curve, defined over the rationals, and allow the base field to be extended to a quadratic number field. How many cases of proper extension in the corresponding torsion groups can we predict to appear? To begin with, for a fixed curve E, then only a small amount of quadratic extensions will be interesting from the point of view of the torsion subgroup. This is a known result; see for instance, [14, Corollary 2] for a different approach or [9, Lemma 3.4] for a similar proof than the one presented here, which we include because it suits our forthcoming arguments. Theorem 6. Let Ebe an elliptic curve defined over the rationals. Then, for all but finitely many quadratic extensions K/Q,E(K)tors =E(Q)tors. Proof. The proof is actually the method used in our computations above. The order of a torsion point defined over a quadratic extension must be m∈ T={1,...,10,12,15,16}. Therefore, for a given curve Edefined over Q, one only has to compute the m–th division polynomials ψm(X)(modd) or ψm(X)/(2Y)(m even) [24], for all m∈ T, and look for irreducible quadratic and linear factors in Q[X]. For quadratic factors, we must consider their splitting fields. For linear factors X−α, we must take the extension (maybe trivial) Q√α3+Aα +B, where E:y2=x3+Ax +B. These extensions, which are obviously finitely many, are the only quadratic ones where the torsion subgroup may grow.  One might in fact give an upper bound for the number of quadratic extensions where E(K)tors 6=E(Q)tors, simply considering the number of linear factors that may appear in ψm(X)or ψm(X)/(2Y), with m∈ T being a prime power. If we want to be more precise we can use Theorem 2 to reduce the number of the division polynomials that must be taken into account. For instance, if E(Q)tors = C3, we know that E(K)tors ∈ {C3,C5,C15,C3×C3}, hence both torsion subgroups are different if and only if E(K)has either a point of order 5or a non–rational 8 ENRIQUE GONZÁLEZ–JIMÉNEZ AND JOSÉ M. TORNERO point of order 3. As deg(ψ3) = 4 and deg(ψ5) = 12 there are, at most, 16 quadratic extensions where the torsion grows. Corollary 7. For any elliptic curve Edefined over the rational field, G=E(Q)tors. The number of quadratic extensions Kverifying E(K)tors 6=Gis bounded by a constant kGthat only depends on G, and is given by: G kGG kGG kGG kGG kG C180 C443 C70C10 1C2×C438 C2182 C54C8128 C12 1C2×C67 C316 C612 C90C2×C242 C2×C80 The result is by no means accurate. In fact, it is not very complicated to sharpen the bound for odd-order groups, the even-order ones being much less understood (in practical terms). Our experimental data suggests that, in fact, the bound might well be 4quadratic extensions for all cases. Example.– The elliptic curve 30a7 has minimal Weierstrass equation: E:Y2+XY +Y=X3−5334X−150368 and E(Q)tors =C2. Even more: D−5−3−2−10 E(Q(√D))tors C4C6C4C2×C2 Theorem 6 allows us to give a different proof of the following result mentioned by Gouvêa and Mazur [8]. Theorem 8. Given an elliptic curve E, defined over Q, there is a finite amount of quadratic twists ED(Q)such that ED(Q)has points of order greater than 2. The number of such quadratic twists is bounded in terms of G=E(Q)tors as in the previous result. Proof. Let us consider K=Q(√D)and the morphism ω:E(K)−→ ED(Q), given by the composition E(K)Ψ −→ E(Q)×ED(Q)π2 −→ ED(Q), where Ψis the mapping given in Theorem 3. That is, ω(P) = φ(P−σP, 1/√D). It is clear that ker(ω) = E(Q)and, as in Theorem 3, if we consider the long exact sequence 0→E(Q)−→ E(K)ω −→ ED(Q)−→ coker(ω)→0, One can see that, for all P∈ED(Q),[2]P=ω(φ(P, √D)). That is, any non– zero element of coker(ω)has order 2. Now, looking at the finite part, Theorem 6 shows that for almost all D, we have E(Q)tors =E(K)tors, so Img(ω)is trivial, and ED(Q)tors ≃coker(ω), which has only elements of order at most 2. However, some questions arise in this context which we have not yet an answer to. Problem 1.– Let us fix G∈Φ(1). Consider the set of all elliptic curves E, defined over Q, with G=E(Q)tors and, when Evaries in this set, give a sharp bound for the maximal number of quadratic extensions K/Qwith G6=E(K)tors. TORSION OF RATIONAL ELLIPTIC CURVES OVER QUADRATIC FIELDS 9 Problem 2.– Is there a precise (and easy) description of which are the possible extensions K/Qwith E(Q)tors 6=E(K)tors, ideally in terms of some invariant(s) of the curve? Problem 3.– For a given G∈Φ(1), one can see experimentally that not all subsets of ΦQ(2, G)appear when one considers an arbitrary curve Ewith G=E(Q)tors and all quadratic extensions K/Q. Find a precise description of all the combinations that may occur. In connection to the last problem, one can prove that there are curves with stable torsion: Proposition 9. Let G∈Φ(1), with G6=C2n. Then there exists an elliptic curve Esuch that, for all quadratic extensions K/Q, G=E(Q)tors =E(K)tors. Proof. Clearly the groups G=C2ndo not satisfy this property, as there is a quadratic extension where full 2–torsion is achieved. The fact that, for all other groups, there are curves with stable torsion for all quadratic extensions can be checked in Table 2.  Finally, some comments about the other possible strategy mentioned at the beginning of the section. If we fix the field, then a thorough study of possible groups is possible [11] studying the non–cuspidal points of certain modular curves. This technique was in fact the main tool in the search (and hunt) for the most unusual group appearing in ΦQ(2), which is C15. As we mentioned before, it only appears in 4very specific cases [20]. 5. Computations Table 2 is the result of our computations with the curves in the Antwerp– Cremona tables, with conductor less than 300000 (a total of 1887909 elliptic curves) must be read as follows: (1) The first column is G∈Φ(1). (2) The second column is H∈Φ2(Q, G). (3) Columns 3rd to 6th display specific examples: •The third column is the elliptic curve Ewith minimal conductor such that E(Q)tors ≃G,E(K)tors ≃H, with K=Q(√D),D > 0being the integer in the fourth column. •Columns 5th and 6th are analogous, with D < 0. •When these four colums are merged, H=Gand the curve in the cell verifies E(Q)tors =E(K)tors for all quadratic extensions K/Q. (4) Columns 7th and 8th indicate the total amount of curves found verifying the corresponding situation (7th for real extensions, 8th for complex, merged for stable torsion groups). Acknowledgement: The authors would like to thank J. Silverman and N. Elkies for their suggestions and ideas. References [1] Birch, B. J.; Kuyk, W. (eds.): Modular Functions of One Variable IV. Lecture Notes in Mathematics 476. Springer (1975).