On the ubiquity of trivial torsion on elliptic curves

arXiv:1002.0667v1 [math.NT] 3 Feb 2010 ON THE UBIQUITY OF TRIVIAL TORSION ON ELLIPTIC CURVES ENRIQUE GONZ´ ALEZ-JIM´ ENEZ AND JOS´ E M. TORNERO Abstract. The purpose of this paper is to give a down–to–earth proof of the well–known fact that a randomly chosen elliptic curve over the rationals is most likely to have trivial torsion. 1. Introduction Let us consider an elliptic curve E, defined over the rationals and written in short Weierstrass form (1) E:Y2=X3+AX +B, A, B ∈Z. We will use the standard notations for: •∆ = −16(4A3+ 27B2)6= 0, the discriminant of E; •E(Q), the finitely generated abelian group of rational points on E, and • O, the identity element of E(Q). Given P∈E(Q), we will also write as customary [m]Pfor the point resulting after adding mtimes P. The problem of computing the torsion of E(Q) has been solved in a lot of very efficient ways [2, 3, 6], and most computer packages (say Maple-Apecs, PARI/GP,Magma or Sage) calculate the torsion of curves with huge coefficients in very few seconds. The major result which made this possible (along with others, like the Nagell–Lutz Theorem ([18],[15]) or the embedding theorem for good reduction primes (see, for example, [21, VIII.7] or [12, Chap. 5])) was Mazur’s Theorem [16, 17] who listed the fifteen possible torsion groups. In the above papers, it is proved that the possible structures of the torsion group of E(Q) are Z/nZfor n= 2,...,10,12,or Z/2Z×Z/2nZfor n= 1,...,4. Besides, the fifteen of them actually happen as torsion subgroups of elliptic curves. Notice that thanks to the above theorem, the possible prime orders for a torsion point defined over Qare 2,3,5 or 7. Let pbe a prime number and let E[p] be the group of points of order pon E(Q), where Qdenotes an algebraic closure of Q. The action of the Date: February 3, 2010. 1 2 ENRIQUE GONZ ´ ALEZ-JIM´ ENEZ AND JOS´ E M. TORNERO absolute Galois group GQ= Gal(Q/Q) on E[p] defines a mod pGalois representation ρE,p : GQ→Aut(E[p]) ∼ =GL2(Fp). Let Q(E[p]) be the number field generated by the coordinates of the points of E[p]. Therefore, the Galois extension Q(E[p])/Qhas Galois group Gal(Q(E[p])/Q)∼ =ρE,p(GQ). The prime pis called exceptional for Eif ρE,p is not surjective. If Ehas complex multiplication then any odd prime number is excepcional. On the other hand, if Edoes not have complex multiplication then Serre [20] proved that Ehas only finitely many exceptional primes. Duke [4] proved that almost all elliptic curves over Qhave no exceptional primes. More precisely, given an elliptic curve Ein a short Weierstrass form as in (1), the height of the elliptic curve is defined as H(E) = max(|A|3,|B|2). Let Mbe a positive integer, and let CH(M) be the set of elliptic curves Ewith H(E)≤M6. For any prime pdenote by Ep(M) the set of elliptic curves E∈ CH(M) such that pis an excepcional prime for E, and by E(M) the union of Ep(M) for all primes. Actually in both sets the elliptic curves were considered up to Q–isomorphisms. Duke then proved that lim M→∞ |E(M)| |CH(M)|= 0. His proof is based on a version of the Chebotarev density theorem, and uses a two-dimensional large sieve inequality together with results of Deuring, Hurwitz and Masser-W¨ustholz. Duke also conjectured the following fact, later proved by Grant [10] |E(M)| ∼ c√M. Being a bit more precise, Grant showed that, in order to efficiently estimate |E(M)|, only E2(M) and E3(M) had to be actually taken into account. Now recall that there is a tight relationship between exceptional primes and torsion orders, because if there is a point of order p, then pis an exceptional prime [20]. Our aim is then giving a down-to-earth proof of the fact that almost all elliptic curves over Qhave trivial torsion, motivated by Duke’s paper. We will use in order to achieve this the characterization of torsion structures given in [7, 8], Mazur’s Theorem [16, 17]; and a theorem by Schmidt [19] on Thue inequalites. Note that we have used a different height notion, more naive in some sense, but nevertheless better suited for our purposes. Let us change a bit the notation and let us call E(A,B):Y2=X3+AX +B ON THE UBIQUITY OF TRIVIAL TORSION ON ELLIPTIC CURVES 3 and, provided ∆ 6= 0, we will denote by E(A,B)(Q)[m] the group of points P∈E(A,B)(Q) such that [m]P=O. Let us write as well C(M) = {(A, B)∈Z2|∆ = −16(4A3+ 27B2)6= 0,|A|,|B| ≤ M}. Tp(M) = {(A, B)∈ C(M)|E(A,B)(Q)[p]6={O}}. T(M) = [ pprime Tp(M) Our version of Duke’s result is then as follows. Theorem 1. With the notations above, lim M→∞ |T (M)| |C(M)|= 0. The proof will lead to extremely coarse bounds for |Tp(M)|which will be proved unsatisfactory in view of experimental data, which we will display subsequently. 2. Proof of Theorem 1. Recall that the possible prime orders of a torsion point defined over Q are 2,3,5 or 7. We will make extensive use of the parametrizations of curves with a point of prescribed order given in [7, 8, 14]. These results have recently been proved useful in showing new properties of the torsion subgroup (see, for instance [1, 9, 13]). First, note that, for a given Awith |A| ≤ Mthere are, at most, two possible choices for Bsuch that ∆ = 0 (and hence, the corresponding curve E(A,B)is not an elliptic curve). Therefore |C(M)| ≥ (2M+ 1)2−2(2M+ 1) = 4M2−1. Let us recall from [7] that a curve E(A,B)with a point of order 2 must verify that there exist z1, z2∈Zsuch that A=z1−z2 2, B =z1z2. Therefore z1|Band for a chosen z1, both z2and Aare determined. Hence, there is at most one pair in T2(M) for every divisor of B. We need now an estimate for the average order of the function d(x), the number of positive divisors of x. The simplest estimation is, probably, the one that can be found in [11], d(1) + d(2) + ... +d(x)∼xlog(x). Therefore, as Mtends to infinity, |T2(M)| ≤ M X x=1 2d(x) + M X x=1 2d(x) + 2M, 4 ENRIQUE GONZ ´ ALEZ-JIM´ ENEZ AND JOS´ E M. TORNERO taking into account that we need to consider both positive and negative divisors, the cases where x∈ {−M, ..., −1}and the 2Mcurves with B= 0. Hence |T2(M)| ∼ c2Mlog(M), where we can, in fact, take c2= 4. As for points of order 3 we can find in [7] a similar characterization (a bit more complicated this time) based on the existence of z1, z2∈Zsuch that A= 27z4 1+ 6z1z2, B =z2 2−27z6 1. Analogously z1|Aand, once we fix such a divisor, z2is necessarily given by z2=A−27z4 1 6z1 , which implies that again there is at most one pair in T3(M) for every divisor of A. Hence, as Mtends to infinity |T3(M)| ≤ c3Mlog(M), and again c3= 4 suits us. Points of order 5 and 7 need a similar, yet slightly different argument. From [8] we know that if there is a point of order 5 in E(A,B)(Q), then there must exist p, q ∈Zverifying: A=−27(q4−12q3p+ 14q2p2+ 12p3q+p4), B= 54(p2+q2)(q4−18q3p+ 74q2p2+ 18p3q+p4). The first equation is an irreducible Thue equation, hence we can apply the following result by Schmidt: Theorem (Schmidt [19]).– Let F(x, y) be an irreducible binary form of degree r > 3, with integral coefficients. Suppose that not more than s+ 1 coefficients are nonzero. Then the number of solutions of the inequality |F(x, y)| ≤ his, a most, (rs)1/2h2/r 1 + log1/r(h). As for our interests are concerned, this gives a bound for the number of possible (p, q) such that −27(q4−12q3p+ 14q2p2+ 12p3q+p4)≤M. Hence, as every such solution determines at most one pair in T5(M), |T5(M)| ≤ 4√M1 + log1/4(M). A similar result can be applied for points of order 7. The equations which must have a solution are now A=−27k4(p2−pq +q2)(q6+ 5q5p−10q4p2−15q3p3+ 30q2p4−11qp5+p6), B= 54k6(p12 −18p11q+ 117p10q2−354p9q3+ 570p8q4−486p7q5 +273p6q6−222p5q7+ 174p4q8−46p3q9−15p2q10 + 6pq11 +q12). ON THE UBIQUITY OF TRIVIAL TORSION ON ELLIPTIC CURVES 5 either for k= 1 or for k= 1/3. Hence, using the polynomial defining B and with a similar argument as above |T7(M)| ≤ 24 6 √M1 + log1/12(M). Therefore, for all pthere is an absolut constant cp∈Z+such that lim M→∞ |Tp(M)| |C(M)|≤lim M→∞ cpMlog(M) 4M2−1= 0. This proves the theorem. Remark.– It must be noted here that our arguments are counting pairs (A, B). So, in fact, isomorphic curves may appear as separated cases. Both Duke and Grant estimated isomorphism classes (over Q) rather than curves. But this can also be achieved by the arguments above with a little extra work. We will show now that these instances of isomorphic curves are actually negligible as for counting is concerned. First note that if two curves E(A,B)and E(A′,B′)are isomorphic over Q, there must be some u∈Qsuch that A=u4A′and B=u6B′. Hence, there exists some prime lsuch that, say, l4|Aand l6|B(the case l4|A′and l6|B′ is analogous). Let us write, for a fixed prime l Pn(M, l) = {x∈Z+|1≤x≤M, ln|M}, and by Pn(M) the union of Pn(M, l), where lrun the set of prime divisors of M. Then it is clear that |Pn(Mn)| ≤ X l≤M|Pn(Mn, l)|=X l≤MMn ln=X l≤MMn ln+O(1)= =MnPl≤M1 ln+O(M) = MnX lprime 1 ln+O(M) = MnP(n) + O(M), where Pis the prime zeta function (see [5], for instance). So, changing Mn for Mwe get |P4(M)| ≤ P(4)M+O4 √M≃0.0769931M+O4 √M, |P6(M)| ≤ P(6)M+O6 √M≃0.0170701M+O6 √M. Hence, if we are interested in curves up to Q–isomorphism, our bounds for |Tp(M)|are still correct, while we should change |C(M)| ≥ 4M2−1 by |C(M)| ≥ (4 −P(4)P(6)) M2+O6 √M which obviously makes no difference in the result. 6 ENRIQUE GONZ ´ ALEZ-JIM´ ENEZ AND JOS´ E M. TORNERO Remark 1.While all of our boundings for |Tp(M)|are of the form cpMlog(M), computational data show that the actual number of curves on Tp(M) depends heavily on p, as one might predict after the estimation given by Grant [10] for Ep(M), the set of elliptic curves E∈ CH(M) such that pis an excepcional prime for E. In fact, a hands–on Magma program gave us the following output M|T2(M)| |T3(M)| |T5(M)| |T7(M)| 104204,220 507 1 1 1052,484,196 1,935 3 1 10629,430,050 5,873 11 4 107340,334,782 18,387 24 5 These actual figures are quite smaller than the bounds obtained. Acknowledgement. The first author was supported in part by grants MTM 2009-07291 (Ministerio de Educaci´on y Ciencia, Spain) and CCG08- UAM/ESP-3906 (Universidad Auton´oma de Madrid-Comunidad de Madrid, Spain). The second author was supported by grants FQM–218 and P08– FQM–03894 (Junta de Andaluc´ıa) and MTM 2007–66929 (Ministerio de Educaci´on y Ciencia, Spain). References [1] M.A. Bennett; P. Ingram: Torsion subgroups of elliptic curves in short Weierstrass form. Trans. Amer. Math. Soc. 357 (2005) 3325–3337. [2] J. E. Cremona: Algorithms for modular elliptic curves. Cambridge University Press, 1992. [3] D. Doud: A procedure to calculate torsion of elliptic curves over Q. Manuscripta Math. 95 (1998) 463–469. [4] W. Duke: Elliptic curves with no exceptional primes. C.R. Acad. Sci. Paris S´erie I 325 (1997) 813–818. [5] C.-E. Fr¨oberg: On the prime zeta function. BIT 8(1968) 187–202. [6] I. Garc´ıa–Selfa; M.A. Olalla; J.M. 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Universidad Aut´ onoma de Madrid, Departamento de Matem´ aticas and Instituto de Ciencias Matem´ aticas (CSIC-UAM-UC3M-UCM), Madrid, Spain E-mail address:[email protected] Departamento de ´ Algebra, Universidad de Sevilla. P.O. 1160. 41080 Sevilla, Spain. E-mail address:[email protected]