On the ubiquity of trivial torsion on elliptic curves
aa r X i v : . [ m a t h . N T ] F e b ON THE UBIQUITY OF TRIVIAL TORSION ONELLIPTIC CURVES
ENRIQUE GONZ ´ALEZ-JIM´ENEZ AND JOS´E M. TORNERO
Abstract.
The purpose of this paper is to give a down–to–earth proofof the well–known fact that a randomly chosen elliptic curve over therationals is most likely to have trivial torsion. Introduction
Let us consider an elliptic curve E , defined over the rationals and writtenin short Weierstrass form(1) E : Y = X + AX + B, A, B ∈ Z . We will use the standard notations for: • ∆ = − A + 27 B ) = 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 ] P for the pointresulting after adding m times P .The problem of computing the torsion of E ( Q ) has been solved in a lot ofvery efficient ways [2, 3, 6], and most computer packages (say Maple-Apecs , PARI/GP , Magma or Sage ) calculate the torsion of curves with huge coeffi-cients in very few seconds. The major result which made this possible (alongwith others, like the Nagell–Lutz Theorem ([18],[15]) or the embedding the-orem 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 torsiongroups.In the above papers, it is proved that the possible structures of the torsiongroup of E ( Q ) are Z /n Z for n = 2 , . . . , , , or Z / Z × Z / n Z for n = 1 , . . . , . Besides, the fifteen of them actually happen as torsion subgroups of el-liptic curves. Notice that thanks to the above theorem, the possible primeorders for a torsion point defined over Q are 2 , , p be a prime number and let E [ p ] be the group of points of order p on E ( Q ), where Q denotes an algebraic closure of Q . The action of the Date : November 21, 2018. absolute Galois group G Q = Gal( Q / Q ) on E [ p ] defines a mod p Galoisrepresentation ρ E,p : G Q → Aut( E [ p ]) ∼ = GL ( F p ) . Let Q ( E [ p ]) be the number field generated by the coordinates of the pointsof E [ p ]. Therefore, the Galois extension Q ( E [ p ]) / Q has Galois groupGal( Q ( E [ p ]) / Q ) ∼ = ρ E,p (G Q ) . The prime p is called exceptional for E if ρ E,p is not surjective. If E hascomplex multiplication then any odd prime number is excepcional. On theother hand, if E does not have complex multiplication then Serre [20] provedthat E has only finitely many exceptional primes.Duke [4] proved that almost all elliptic curves over Q have no exceptionalprimes. More precisely, given an elliptic curve E in a short Weierstrass formas in (1), the height of the elliptic curve is defined as H ( E ) = max( | A | , | B | ) . Let M be a positive integer, and let C H ( M ) be the set of elliptic curves E with H ( E ) ≤ M . For any prime p denote by E p ( M ) the set of ellipticcurves E ∈ C H ( M ) such that p is an excepcional prime for E , and by E ( M )the union of E p ( M ) for all primes. Actually in both sets the elliptic curveswere considered up to Q –isomorphisms. Duke then proved thatlim M →∞ |E ( M ) ||C H ( M ) | = 0 . His proof is based on a version of the Chebotarev density theorem, and usesa 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 E ( M ) and E ( M ) had to be actually taken into account.Now recall that there is a tight relationship between exceptional primesand torsion orders, because if there is a point of order p , then p is anexceptional prime [20]. Our aim is then giving a down-to-earth proof of thefact that almost all elliptic curves over Q have trivial torsion, motivated byDuke’s paper.We will use in order to achieve this the characterization of torsion struc-tures 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 ) : Y = X + AX + B N THE UBIQUITY OF TRIVIAL TORSION ON ELLIPTIC CURVES 3 and, provided ∆ = 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 ) ∈ Z | ∆ = − A + 27 B ) = 0 , | A | , | B | ≤ M } . T p ( M ) = { ( A, B ) ∈ C ( M ) | E ( A,B ) ( Q )[ p ] = {O}} . T ( M ) = [ p prime T p ( 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 |T p ( M ) | which will beproved unsatisfactory in view of experimental data, which we will displaysubsequently. 2. Proof of Theorem 1.
Recall that the possible prime orders of a torsion point defined over Q are 2 , , A with | A | ≤ M there are, at most, twopossible choices for B such that ∆ = 0 (and hence, the corresponding curve E ( A,B ) is not an elliptic curve). Therefore |C ( M ) | ≥ (2 M + 1) − M + 1) = 4 M − . Let us recall from [7] that a curve E ( A,B ) with a point of order 2 mustverify that there exist z , z ∈ Z such that A = z − z , B = z z . Therefore z | B and for a chosen z , both z and A are determined. Hence,there is at most one pair in T ( M ) for every divisor of B .We need now an estimate for the average order of the function d ( x ), thenumber of positive divisors of x . The simplest estimation is, probably, theone that can be found in [11], d (1) + d (2) + ... + d ( x ) ∼ x log( x ) . Therefore, as M tends to infinity, |T ( M ) | ≤ M X x =1 d ( x ) + M X x =1 d ( x ) + 2 M, ENRIQUE GONZ ´ALEZ-JIM´ENEZ AND JOS´E M. TORNERO taking into account that we need to consider both positive and negativedivisors, the cases where x ∈ {− M, ..., − } and the 2 M curves with B = 0.Hence |T ( M ) | ∼ c M log( M ), where we can, in fact, take c = 4.As for points of order 3 we can find in [7] a similar characterization (a bitmore complicated this time) based on the existence of z , z ∈ Z such that A = 27 z + 6 z z , B = z − z . Analogously z | A and, once we fix such a divisor, z is necessarily given by z = A − z z , which implies that again there is at most one pair in T ( M ) for every divisorof A . Hence, as M tends to infinity |T ( M ) | ≤ c M log( M ) , and again c = 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 theremust exist p, q ∈ Z verifying: A = − q − q p + 14 q p + 12 p q + p ) ,B = 54( p + q )( q − q p + 74 q p + 18 p q + p ) . The first equation is an irreducible Thue equation, hence we can applythe following result by Schmidt:
Theorem (Schmidt [19] ).–
Let F ( x, y ) be an irreducible binary form ofdegree r >
3, with integral coefficients. Suppose that not more than s + 1coefficients are nonzero. Then the number of solutions of the inequality | F ( x, y ) | ≤ h is, a most,( rs ) / h /r (cid:16) /r ( h ) (cid:17) . As for our interests are concerned, this gives a bound for the number ofpossible ( p, q ) such that (cid:12)(cid:12) − q − q p + 14 q p + 12 p q + p ) (cid:12)(cid:12) ≤ M. Hence, as every such solution determines at most one pair in T ( M ), |T ( M ) | ≤ √ M (cid:16) / ( M ) (cid:17) . A similar result can be applied for points of order 7. The equations whichmust have a solution are now A = − k ( p − pq + q )( q + 5 q p − q p − q p +30 q p − qp + p ) ,B = 54 k ( p − p q + 117 p q − p q + 570 p q − p q +273 p q − p q + 174 p q − p q − p q + 6 pq + q ) . N 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 |T ( M ) | ≤ √ M (cid:16) / ( M ) (cid:17) . Therefore, for all p there is an absolut constant c p ∈ Z + such thatlim M →∞ |T p ( M ) || C ( M ) | ≤ lim M →∞ c p M log( M )4 M − . 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. BothDuke and Grant estimated isomorphism classes (over Q ) rather than curves.But this can also be achieved by the arguments above with a little extrawork. We will show now that these instances of isomorphic curves areactually 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 ∈ Q such that A = u A ′ and B = u B ′ . Hence, thereexists some prime l such that, say, l | A and l | B (the case l | A ′ and l | B ′ is analogous). Let us write, for a fixed prime lP n ( M, l ) = { x ∈ Z + | ≤ x ≤ M, l n | M } , and by P n ( M ) the union of P n ( M, l ), where l run the set of prime divisorsof M .Then it is clear that | P n ( M n ) | ≤ X l ≤ M | P n ( M n , l ) | = X l ≤ M (cid:20) M n l n (cid:21) = X l ≤ M (cid:18) M n l n + O (1) (cid:19) == M n P l ≤ M (cid:0) l n (cid:1) + O ( M ) = M n X l prime l n + O ( M ) = M n P ( n ) + O ( M ) , where P is the prime zeta function (see [5], for instance). So, changing M n for M we get | P ( M ) | ≤ P (4) M + O (cid:16) √ M (cid:17) ≃ . M + O (cid:16) √ M (cid:17) , | P ( M ) | ≤ P (6) M + O (cid:16) √ M (cid:17) ≃ . M + O (cid:16) √ M (cid:17) . Hence, if we are interested in curves up to Q –isomorphism, our boundsfor |T p ( M ) | are still correct, while we should change |C ( M ) | ≥ M − |C ( M ) | ≥ (4 − P (4) P (6)) M + O (cid:16) √ M (cid:17) which obviously makes no difference in the result. ENRIQUE GONZ ´ALEZ-JIM´ENEZ AND JOS´E M. TORNERO
Remark . While all of our boundings for |T p ( M ) | are of the form c p M log( M ),computational data show that the actual number of curves on T p ( M ) de-pends heavily on p , as one might predict after the estimation given by Grant[10] for E p ( M ), the set of elliptic curves E ∈ C H ( M ) such that p is an excep-cional prime for E . In fact, a hands–on Magma program gave us the followingoutput M |T ( M ) | |T ( M ) | |T ( M ) | |T ( M ) | ,
220 507 1 110 , ,
196 1 ,
935 3 110 , ,
050 5 ,
873 11 410 , ,
782 18 ,
387 24 5These actual figures are quite smaller than the bounds obtained.
Acknowledgement.
The first author was supported in part by grantsMTM 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 deEducaci´on y Ciencia, Spain).
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