Uniform bounds on the number of rational points of a family of curves of genus 2

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Uniform bounds on the number of rational points of a family of curves of genus 2$ L. Kulesz,a G. Matera,a,c,
and E. Schostb a

Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Campus Universitario, Jose´ M. Gutie´rrez 1150 (1613) Los Polvorines, Buenos Aires, Argentina b Laboratoire GAGE, E´cole Polytechnique, F-91128 Palaiseau Cedex, France c Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas (CONICET), Argentina Received 7 November 2003; revised 12 March 2004 Available online 27 July 2004 Communicated by B. Poonen

Abstract We exhibit a genus-2 curve C defined over QðTÞ which admits two independent morphisms to a rank-1 elliptic curve defined over QðTÞ: We describe completely the set of QðTÞ-rational points of the curve C and obtain a uniform bound on the number of Q-rational points of a rational specialization Ct of the curve C for a certain (possibly infinite) set of values tAQ: Furthermore, for this set of values tAQ we describe completely the set of Q-rational points of the curve Ct : Finally, we show how these results can be strengthened assuming a height conjecture of Lang. r 2004 Elsevier Inc. All rights reserved. Keywords: Genus 2-curves; Elliptic curves; Rational points; Demj’anenko–Manin’s method; Specialization morphisms

$ Research was partially supported by the following Argentinian and French Grants: UBACyT X198, PIP CONICET 2461, ECOS A99E06, UNGS 30/3005. Some of the results presented here were first announced at the Workshop Argentino de Informa´tica Teo´rica, WAIT’01, held in September 2001 (see [10]).
Corresponding author. Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Campus Universitario, Jose´ M. Gutie´rrez 1150 (1613) Los Polvorines, Buenos Aires, Argentina. E-mail addresses: [email protected] (L. Kulesz), [email protected] (G. Matera), Eric.Schost@ polytechnique.fr (E. Schost). URLs: http://www.medicis.polytechnique.fr/~matera, http://www.stix.polytechnique.fr/~schost.

0022-314X/$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2004.05.013

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1. Introduction In 1983, Faltings proved Mordell’s Conjecture, which asserts that for any number field K; the set CðKÞ of K-rational points of a curve C defined over K of genus at least 2 is finite (see [8]). In order to have more insight on Faltings’ Theorem one may ask about the behaviour of the set of K-rational points of a given K-definable family f : S-P1 ðQÞ of curves of (fixed) genus X2: This question is strongly related to the following conjecture of Lang [18]: Conjecture A. Let V be a variety of general type defined over a number field K: Then the set V ðKÞ of K-rational points of V is contained in a subvariety of V of codimension at least 1. As an attempt to understand Conjecture A, Caporaso et al. [1,2] showed the following consequence of this conjecture in the case of algebraic curves: Theorem 1. If Conjecture A is true, then for any number field K and any integer gX2 there exists an integer BðK; gÞ such that any non-singular curve defined over K of genus g has at most BðK; gÞ K-rational points. Partial results in the direction of Theorem 1, namely uniform upper bounds on the number of Q-rational points of families of curves of genus X2; were obtained in [16,32,34,37]. These articles consider families of twists of certain fixed curves of genus X2 and a family of curves defined by a Thue’s equation. In this article we obtain uniform upper bounds on the number of Q-rational points of the (non-isotrivial) family of plane curves fCt gtAQ of equation y2 ¼ x6 þ tx4 þ tx2 þ 1; for the (possibly infinite) set of values tAQ for which the group E t ðQÞ of Q-rational points of the elliptic curve E t of equation y2 ¼ x3 þ tx2 þ tx þ 1 has rank 1 and its free part is generated by ð0; 1Þ: By means of a direct computation of the invariants of the curve Ct we see that for all but finitely many pairs ðt; uÞAQ2 with tau the curves Ct and Cu are isomorphic over C if and only if u ¼ 15 t 1þt holds. Furthermore, this isomorphism is Q-definable if and only if 2 þ 2t is a square in Q: This implies that the family of curves fCt gtAQ contains infinitely many non-Q-isomorphic curves. Let us observe that the family of curves fCt gtAQ may be intrinsically defined in the following terms: it is (up to Q-isomorphism) the only family of genus-2 curves with two independent degree-2 morphisms to a family of elliptic curves with a distinguished rational 2-torsion point. Indeed, following e.g. [4] we see that any Q-definable genus-2 curve with a degree-2 morphism to an elliptic curve is isomorphic to a curve Ca;b of equation

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y2 ¼ x6 þ ax4 þ bx2 þ 1 for suitable a; bAQ: This implies that the curve Ca;b admits two independent degree-2 morphisms to the elliptic curves of equations y2 ¼ x3 þ ax2 þ bx þ 1 and y2 ¼ x3 þ bx2 þ ax þ 1: Let lAQ be such that l2 þ l þ 1 ¼ 0: Then the above elliptic curves have the same j-invariant if and only if one of the following conditions hold: (i) b ¼ a; (ii) b ¼ a 3; (iii) b ¼ la or b ¼ ðl þ 1Þa; (iv) b ¼ la þ 3ðl þ 1Þ or b ¼ ðl þ 1Þa 3l: A direct computation shows that the unidimensional family of curves fCa; b gaAQ corresponding to cases (iii) and (iv) is Q-isomorphic to one of the families corresponding to cases (i) and (ii). On the other hand, the family of curves corresponding to case (ii) is mapped into the families of elliptic curves fE a;1 gaAQ ; fE a;2 gaAQ of equation y2 ¼ x3 þ ax2 þ ax þ 1 and y2 ¼ x3 þ ax2 ða þ 3Þx þ 1; respectively. Since E a;2 does not have any 2-torsion point defined over QðaÞ we conclude that the family fCt gtAQ ; which corresponds to the case (i), is characterized by the property of having two independent degree-2 morphism to one family of elliptic curves with a distinguished rational 2-torsion point. Let T denote an indeterminate over Q; let QðTÞ and QðTÞ denote the field of rational functions in the variable T with coefficients in Q and Q; respectively, and let QðTÞ denote the algebraic closure of QðTÞ: First, we analyse the arithmetic behaviour of the plane curve C defined over QðTÞ of equation y2 ¼ x6 þ Tx4 þ Tx2 þ 1: Our methods rely on the observation that the (independent) morphisms f1 ; f2 defined by
 1 y 2 f1 ðx; yÞ :¼ ðx ; yÞ; f2 ðx; yÞ :¼ ; ; x2 x3 map the curve C into the elliptic curve E defined over QðTÞ of equation y2 ¼ x3 þ Tx2 þ Tx þ 1: First, applying Shioda’s theory of Mordell–Weil lattices, we prove that the group of QðTÞ-rational points of E has rank one and its torsion subgroup is given by EðQðTÞÞtors ¼ fOE ; ð 1; 0Þg: Then, using a variant of Dem’janenko–Manin’s method [7,20] to find the set of rational points of a given plane curve, we obtain the following result: Theorem 2. CðQðTÞÞ ¼ fð0; 1Þ; ð0; 1Þg: Then for a given value tAQ we analyse the arithmetic behaviour of the curve Ct using Dem’janenko–Manin’s method. For this purpose, we observe that 2 ; f2 j 2 of the morphisms f1 ; f2 defined above map the the restriction f1 j C-Q

C-Q

curve Ct into the elliptic curve E t defined over Q of equation y2 ¼ x3 þ tx2 þ tx þ 1: For any value tAQ such that the abelian group E t ðQÞ of Q-rational points of the elliptic curve E t has rank 1 and its free part is generated by the point ð0; 1Þ; we determine the set Ct ðQÞ of Q-rational points of the curve Ct : We prove the following result:

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Theorem 3. Let PCQ denote the set of values tAQ for which the abelian group E t ðQÞ has rank 1 and its free part is generated by the point (0,1). Then the following statements hold for all but finitely many tAP: (i) If there exists vAQ such that t ¼ ðv4 v2 þ 1Þ=v2 holds, then 

 1 1 Ct ðQÞ ¼ ð0; 1Þ; ð0; 1Þ; ðv; 0Þ; ð v; 0Þ; ; 0 ; ; 0 : v v (ii) Otherwise, we have Ct ðQÞ ¼ fð0; 1Þ; ð0; 1Þg: Let h and hˆ denote the naive (logarithmic) height on Q and the canonical height on e defined over Q; respectively (see the next section for precise a given elliptic curve E definitions). Then the statement of Theorem 3 can be significantly improved, assuming that the following conjecture of Lang holds [17]: e Conjecture B. There exists a universal constant c40 such that for any elliptic curve E e defined over Q of discriminant D and any non-torsion point PAE ðQÞ; the estimate ˆ hðPÞ4c  hðDÞ holds. Let us observe that Conjecture B has been proved for elliptic curves with integral j-invariant [35]. Furthermore, [11] shows that the abc-conjecture implies Conjecture B. Under the assumption of the validity of Conjecture B we have the following result, which shows that the condition that ð0; 1Þ is a generator of the free part of the group E t ðQÞ is not essential: Theorem 4. If Conjecture B is true there exists a universal constant C40 with the following property: for any tAQ such that the abelian group E t ðQÞ has rank 1, the cardinality of the set Ct ðQÞ is bounded by C: Finally, let us observe that the validity of the statement of Theorems 3 and 4 depends on either or both of the following conditions on the parameter tAQ: 1. The rank of the abelian group E t ðQÞ is 1. 2. (0,1) is a generator of the free part of E t ðQÞ: In Section 5 we discuss how restrictive these conditions on the parameter tAQ are. Theorem 4 shows that our uniform upper bound on the cardinality of the set Ct ðQÞ does not depend on condition 2 if Conjecture B holds. We exhibit statistical results which seem to show that condition 1 holds with a probability of success of approximately 1=3: Furthermore, let Q be the set of values tAQ for which E t ðQÞ has

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rank 1. Our experimental results seem to show that the set of values tAQ for which (0,1) is a generator of the free part of E t ðQÞ has density 1 in Q: The results of this article required an important computational effort. The experimental results presented in Section 5 were done using Cremona’s software mwrank [6] and took about 2 months of CPU time on a 1 GHz PC. All the other symbolic computations were done using the Magma computer algebra system [19]. All software and hardware resources were provided by the French computation center MEDICIS [22].

2. Basic notions and results In this section we fix notations and recall some standard notions and results about elliptic curves, heights and morphisms. Details can be found in [15,31,35]. Let K denote any of the fields Q or QðTÞ and let OK denote its ring of integers i.e. Z or Q½T; respectively. For x ¼ x1 =x2 AK with x1 AOK ; x2 AO
K and gcdðx1 ; x2 Þ ¼ 1; we denote by hðxÞ the (naive) height of x; namely hðxÞ :¼ logðmaxfjx1 j; jx2 jgÞ if K ¼ Q and hðxÞ :¼ maxfdegðx1 Þ; degðx2 Þg if K ¼ QðTÞ: For a given algebraic curve C defined over K we denote by CðKÞ the set of points of the curve C whose coordinates lie in K: % of equation y2 ¼ Let C be the K-definable affine (hyperelliptic) curve of A2 ðKÞ f ðxÞ; where f AK½x is a square-free polynomial of degree at least 3. For any point P ¼ ðxðPÞ; yðPÞÞACðKÞ we define the (naive) height hðPÞ of P as hðPÞ :¼ hðxðPÞÞ: % is the point of C at infinity we define hðPÞ :¼ 0: Further, if PAP2 ðKÞ Let E be an elliptic curve defined over K and let ½n denote the morphism of ˆ multiplication by n in E for any nAZ\f0g: For any point PAEðKÞ we denote by hðPÞ

n n ˆ % the canonical height of P; namely hðPÞ :¼ limn-N 4 hð½2 PÞ: For P; QAEðKÞ let ˆ þ QÞ hðPÞ ˆ

/P; QS denote the Ne´ron–Tate pairing, namely /P; QS :¼ 12ðhðP ˆ hðQÞÞ: Let us recall that /; S induces a positive-definite bilinear form on EðKÞ=EðKÞtors ; where EðKÞtors denote the set of K-rational points of torsion of E: It is well-known (see e.g. [31, Theorem 9.3]) that the difference between the canonical and the naive height is uniformly bounded on any given elliptic curve E defined over K; i.e. there exists a universal constant cE 40; depending only on the elliptic curve E; such that the estimate ˆ jhðPÞ

hðPÞjocE

ð1Þ

holds for any PAEðKÞ: The following result will allow us to make the constant cE explicit (see e.g. [15]): Lemma 1. Let E be an elliptic curve defined over K and let cE 40 be a constant satisfying the inequality jhð½2PÞ 4hðPÞjpcE for any point PAEðKÞ: Then the ˆ inequality jhðPÞ

hðPÞjpcE =3 holds for any point PAEðKÞ:

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3. Points over QðTÞ This section is devoted to the proof of Theorem 2, which determines the set of QðTÞ-rational points of the genus-2 curve C of equation y2 ¼ x6 þ Tx4 þ Tx2 þ 1: As expressed in the introduction, there are two QðTÞ-definable morphisms f1 ; f2 : C-E mapping this curve to the elliptic curve E defined over QðTÞ of equation y2 ¼ x3 þ Tx2 þ Tx þ 1: In order to determine the set CðQðTÞÞ we first determine the structure of the group EðQðTÞÞ: 3.1. The structure of E over QðTÞ In order to analyse the group EðQðTÞÞ we need an explicit upper bound on the difference between the canonical and naive height on E: Our next result yields such an upper bound for a short Weierstrass form of E: More precisely, making the change of variable x0 ¼ x þ T=3 we transform the elliptic curve E into the elliptic curve E 0 defined over QðTÞ of equation y2 ¼ x03 þ a0 x0 þ b0 ; where a0 :¼ 1=3TðT 3Þ and b0 :¼ 1=27ð2T þ 3ÞðT 3Þ2 : Then we have the following result: Lemma 2. Let notations and assumptions be as above. Then for any rational point ˆ PAE 0 ðQðTÞÞ the inequality jhðPÞ

hðPÞjp3=4 holds. Proof. Following [39], let MQðTÞ denote the usual set of all non-equivalent absolute values over QðTÞ; namely the set of all the absolute values vp :¼ logj jp ; where either p ¼ N and jF jp :¼ edeg ðF Þ; or p runs over the set of all monic prime elements of Q½T; and jF jp :¼ e ordp ðF Þ denotes the standard p-adic valuation. For any such absolute value v; let 

 1 0 1 0 mv :¼ min vða Þ; vðb Þ ; 2 3 ml :¼

1 X 2 vAM

QðTÞ

minf0; mv g;

m :¼

X

mv ;

vAMQðTÞ

mu :¼

1 2

X

maxf0; mv g:

vAMQðTÞ

ˆ

hðPÞp ml Then [39, Theorem and Proposition 4] shows that m mu phðPÞ 0 holds for any PAE ðQðTÞÞ: In our case, the only non-zero values of mv are obtained at p ¼ N and T 3; namely mN ¼ 1 and mT 3 ¼ 1=2: This shows that m ¼ 1=2; ml ¼ 1=2 and mu ¼ 1=4 ˆ hold, and then 3=4phðPÞ

hðPÞp1=2: This proves the lemma. & Now we determine the structure of the group of QðTÞ-rational points of the elliptic curve E: For this purpose, we are going to apply Shioda’s theory of

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Mordell–Weil lattices of elliptic surfaces (cf. [23,26,27]), which actually allows us to describe the larger group EðQðTÞÞ: Following [26], associated to the elliptic curve E we have an elliptic surface f : S-P1 ðQÞ (the Kodaira–Ne´ron model of E=QðTÞ) whose generic fibre is E: For a given vAP1 ðQÞ let Fv :¼ f 1 ðvÞ denote the fibre over v; and let R denote the set of reducible fibres Fv : For any vAR; let Fv ¼ Yv;0 þ

m v 1 X

mv;i Yv;i ;

i¼1

where Yv;i ð0pipmv 1Þ are the irreducible components of Fv occurring with multiplicity mv;i and Yv;0 is the unique component meeting the zero section. The global sections of S can be naturally identified with the points of EðQðTÞÞ; namely a given section s : P1 ðQÞ-S is identified with its restriction to the generic fibre E; which is a QðTÞ-rational point of E: For a given point PAEðQðTÞÞ let ðPÞ denote the prime divisor which is the image of the section P : P1 ðQÞ-S: With this identification Shioda shows that EðQðTÞÞ is isomorphic to NSðSÞ=T; where NSðSÞ denotes the Ne´ron–Severi group of S (the group of divisors of S modulo algebraic equivalence) and T denotes the subgroup of NSðSÞ generated by the zero section ðOÞ and all the irreducible components of fibres. In [23] there is a complete classification of the possible structures of the group EðQðTÞÞ in terms of the root lattices associated with the reducible fibres Fv : There exists a height pairing /; S : EðQðTÞÞ  EðQðTÞÞ-Q; which is obtained by embedding EðQðTÞÞ into NSðSÞ#Q: Let us denote by f this embedding. Then we have ker f ¼ EðQðTÞÞtors ; and using the intersection number as a pairing in NSðSÞ the height pairing is defined by /P; QS :¼ ðfðPÞ; fðQÞÞ: In case that the elliptic surface is rational we have /P; PS ¼ 2 þ ððPÞ; OÞ

X

contrv ðPÞ;

ð2Þ

vAR

where the possible terms contrv ðPÞ are described in [26] in terms of the root lattice associated to the fibre Fv : Proposition 1. The rank of the abelian group EðQðTÞÞ is one and its free part is generated by the point G :¼ ð0; 1Þ: Proof. Let us observe that the singular fibres of S are given at v ¼ 1; 3; N: By applying Tate’s algorithm for the determination of the reduction types of the fibre Fv (see [35,38]) we see that the special fibres at v ¼ 1; 3; N are of type I1 ; III, I
2 ; respectively. This implies m 1 ¼ 1; m3 ¼ 2 and mN ¼ 7; respectively. Therefore, only v ¼ 3; N correspond to reducible fibres. Applying the classification of [23] we

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conclude that EðQðTÞÞDA
1 "Z=2Z holds, i.e. EðQðTÞÞ has rank 1 and EðQðTÞÞtors ¼ Z=2Z: Since ð 1; 0Þ is a non-trivial torsion point of EðQðTÞÞ we conclude that the identity EðQðTÞÞtors ¼ /ð 1; 0ÞS holds. Let us observe that the elliptic surface associated to the elliptic curve E is rational. Therefore, [26, Theorems 10.8 and 10.10] shows that the group EðQðTÞÞ is generated by the points P ¼ ðxðPÞ; yðPÞÞ satisfying ððPÞ; OÞ ¼ 0; and hence of the form xðPÞ ¼ gT 2 þ aT þ b; yðPÞ ¼ hT 3 þ cT 2 þ dT þ e: From [26, Lemma 5.1] we see that A
1 has a basis consisting of a vector P of (minimal) norm /P; PS ¼ 1=2: Taking into account that contrN ðPÞAf0; 1; 3=2g and contr3 ðPÞAf0; 1=2g holds (see [26]), from formula (2) we conclude that contrN ðPÞa0 holds. Arguing as in [28] we see that this implies that P must intersect the singular fibre FN (which is a cusp) at the singular point, namely at (0,0). We conclude that g ¼ h ¼ 0 holds. Replacing xðPÞ ¼ aT þ b in the right-hand term of the equation defining the elliptic curve E we see that the term pa;b ðTÞ :¼ ðaT þ bÞ3 þ TðaT þ bÞ2 þ TðaT þ bÞ þ 1 is not a square in Q½T for aa0; 1 because it has odd degree. For a ¼ 1 and any b we have that Pa;b is not a square in Q½T: Furthermore, for ba0; 1 the polynomial p0;b ðTÞ ¼ Tðb2 þ bÞ þ b3 þ 1 is not a square. Since b ¼ 1 yields a torsion point we conclude that a ¼ b ¼ 0 is the only possible choice for xðPÞ: This shows that G ¼ ð0; 71Þ is a generator of the free part of EðQðTÞÞ: &

3.2. The structure of C over QðTÞ: proof of Theorem 2 In this section we prove the following result: Theorem 2. Let C be the genus-2 plane curve C defined over QðTÞ of equation y2 ¼ x6 þ Tx4 þ Tx2 þ 1: Then we have CðQðTÞÞ ¼ fð0; 1Þ; ð0; 1Þg: For this purpose we are going to use a simplified version of the Dem’janenko– Manin’s method [7,20] for computing the set of rational points of a given genus-2 curve. Proof. Let us recall that we have two morphisms f1 ; f2 : C-E mapping the curve C into the elliptic curve E; namely f1 ðx; yÞ :¼ ðx2 ; yÞ and f2 ðx; yÞ :¼ ð1=x2 ; y=x3 Þ: As in the proof of Lemma 2 we make the change of variable x0 ¼ x þ T=3; which transforms the elliptic curve E into the elliptic curve E 0 of equation y2 ¼ x03 þ a0 x0 þ b0 ; where a0 :¼ 1=3TðT 3Þ and b0 :¼ 1=27ð2T þ 3ÞðT 3Þ2 : We denote by C0 the genus-2 curve defined over QðTÞ obtained from C under this change of variables and denote by f01 ; f02 : C0 -E 0 the corresponding morphisms,

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namely f01 ðx0 ; yÞ :¼ ððx0 T=3Þ2 þ T=3; yÞ; f02 ðx0 ; yÞ :¼ ððx0 T=3Þ 2 þ T=3; yðx0 T=3Þ 3 Þ: We claim that for any PAC0 ðQðTÞÞ the following inequality holds: jhðf01 ðPÞÞ hðf02 ðPÞÞjp1:

ð3Þ

Indeed, let P be an arbitrary element of C0 ðQðTÞÞ and let x0 ðPÞ ¼ N=D be a reduced representation of x0 ðPÞ: Then the abscissa of f01 ðPÞ is ðð3N DTÞ2 þ 3TD2 Þ=ð9D2 Þ: Observe that ðð3N DTÞ2 þ 3TD2 Þ=ð9D2 Þ is a reduced fraction and hence hðf1 0 ðPÞÞ ¼ maxfdegðð3N DTÞ2 þ 3TD2 Þ; degð9D2 Þg holds. Since ð3N DTÞ2 and 3TD2 have positive leading coefficients, we conclude that degðð3N DTÞ2 þ holds and then 3TD2 ÞÞ ¼ maxfdegðð3N DTÞ2 Þ; degð3TD2 Þg4degð9D2 Þ 2 0 2 hðf1 ðPÞÞ ¼ maxfdegðð3N DTÞ Þ; degð3TD Þg: Similarly, we see that the abscissa of f02 ðPÞ is ð27D2 þ Tð3N DTÞ2 Þ=ð3ð3N DTÞ2 Þ and hðf02 ðPÞÞ ¼ maxfdegð27D2 Þ; degðTð3N DTÞ2 Þg holds. Let a :¼ degðDÞ; b :¼ degð3N DTÞ: We have hðf01 ðPÞÞ ¼ maxf2a þ 1; 2bg and hðf02 ðPÞÞ ¼ maxf2a; 2b þ 1g; which immediately implies estimate (3). This completes the proof of our claim. Proposition 1 asserts that the abelian group E 0 ðQðTÞÞ has rank 1 and G0 :¼ ðT=3; 1Þ is a generator of its free part. Then for any point PAC0 ðQðTÞÞ there exist integers n; m and points T 1 ; T 2 AE 0 ðQðTÞÞtors satisfying the identities f01 ðPÞ ¼ ½nG 0 þ T 1 and f02 ðPÞ ¼ ½mG 0 þ T 2 : Then we have ˆ 0 ðPÞÞ ¼ n2 hðG ˆ 0 Þ; hðf 1

ˆ 0 Þ: ˆ 0 ðPÞÞ ¼ m2 hðG hðf 2

ð4Þ

Hence, combining identity (3) and Lemma 2 we obtain the following estimate: ˆ 0 ðPÞÞ hðf ˆ 0 ðPÞÞjp jhðf ˆ 0 ðPÞÞ hðf0 ðPÞÞj þ jhðf ˆ 0 ðPÞÞ hðf0 ðPÞÞj jhðf 1 2 1 1 2 2 þ jhðf01 ðPÞÞ hðf02 ðPÞÞj p 2  3=4 þ 1 ¼ 5=2:

ð5Þ

Let us suppose first that f01 ðPÞ7f02 ðPÞeE 0 ðQðTÞÞtors holds. Then m2 n2 a0 and ˆ 0 Þjm2 n2 jo5=2: Taking into account that hð½5G 0 Þ ¼ 15 Eqs. (4) and (5) imply hðG ˆ 0 ÞX1=2: Therefore, we have holds, from Lemma 2 we obtain the estimate hðG minfjnj; jmjgo5=2 and hence n; mAf0; 71; 72g:

ð6Þ

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A direct computation shows that the only QðTÞ-rational points of C0 satisfying the condition f01 ðPÞ7f02 ðPÞeE 0 ðQðTÞÞtors are fðT=3; 1Þ; ðT=3; 1Þg: We conclude that the only QðTÞ-rational points of C satisfying the condition f1 ðPÞ7f2 ðPÞeEðQðTÞÞtors are fð0; 1Þ; ð0; 1Þg: On the other hand, suppose that f1 ðPÞ7f2 ðPÞAEðQðTÞÞtors ¼ fOE ; ð 1; 0Þg holds, where OE denotes the zero element of the group EðQðTÞÞ: We have that ðf1 þ f2 Þðx; yÞ ¼ ð fþ ðxÞ; ygþ ðxÞÞ and ðf1 f2 Þðx; yÞ ¼ ð f ðxÞ; yg ðxÞÞ; where fþ ðxÞ ¼

2x3 3x2 2x þ Tx2 ; ðx4 þ 2x3 þ 2x2 þ 2x þ 1Þ

f ðxÞ ¼

2x3 3x2 þ 2x þ Tx2 : ðx4 2x3 þ 2x2 2x þ 1Þ

From the expression of fþ and f we easily conclude that there do not exist points PACðQðTÞÞ for which f1 ðPÞ7f2 ðPÞAfOE ; ð 1; 0Þg holds. Therefore, the image of the morphisms f1 ; f2 is contained in the set fð0; 1Þ; ð0; 1Þg: In particular we see that xðPÞ ¼ 0 holds for any point PACðQðTÞÞ: This shows that CðQðTÞÞ ¼ fð0; 1Þ; ð0; 1Þg and completes the proof of Theorem 2. &

4. Points over Q Let tAQ and let Ct be the curve of equation y2 ¼ x6 þ tx4 þ tx2 þ 1: The purpose of this section is to analyse the arithmetic structure of the curve Ct : For this purpose we first determine the arithmetic structure of the elliptic curve E t of equation y2 ¼ x3 þ tx2 þ tx þ 1: 4.1. Explicit bounds In this section we obtain an explicit upper bound on the height hðPÞ of any point PAE t ðQÞ in terms of the height of t: For this purpose, we first obtain an explicit upper bound on the difference between the naive and the canonical height on E t : Let us observe that general estimates on the difference between the naive and the canonical height were already given in e.g. [33,39]. Nevertheless, the following explicit estimate gives better bounds in this case, which allows us to significantly reduce the subsequent computational effort. Lemma 3. Let tAQ: Then for any Q-rational point P of the elliptic curve E t the following estimate holds: ˆ jhðPÞ

hðPÞjp

5hðtÞ þ logð1314Þ : 3

Proof. Let t :¼ b=a and let P be a point of E b=a ðQÞ: Let us suppose first that P is not a 2-torsion point. This implies that xðPÞ does not cancel the 2-division polynomial x3 þ ðb=aÞx2 þ ðb=aÞx þ 1: Then the x-coordinate of the point ½2P is given by the

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expression xð½2PÞ ¼

a2 xðPÞ4 2abxðPÞ2 8a2 xðPÞ 4ab þ b2 4aðaxðPÞ3 þ bxðPÞ2 þ bxðPÞ þ aÞ

:

ð7Þ

Let us write xðPÞ :¼ p=q; where p and q are coprime integers. Then we have hðPÞ ¼ maxflog jpj; log jqjg: Rewriting identity (7) in terms of p and q we obtain xð½2PÞ ¼

a2 p4 2abp2 q2 8a2 pq3 þ ðb2 4abÞq4 : 4qaðap3 þ bp2 q þ bpq2 þ aq3 Þ

Let N :¼ a2 p4 2abp2 q2 8a2 pq3 þ ðb2 4abÞq4 and D :¼ 4qaðap3 þ bp2 q þ bpq2 þ aq3 Þ denote the numerator and denominator of the above expression. Then we have the estimates jNjp ðjaj2 þ 2jabj þ 8jaj2 þ jb2 4abjÞ maxfjpj; jqjg4 p 16 maxfjaj; jbjg2 maxfjpj; jqjg4 ; jDjp 4ðjaj2 þ jbaj þ jbaj þ jaj2 Þ maxfjpj; jqjg4 p 16 maxfjaj; jbjg2 maxfjpj; jqjg4 : This yields hðxð½2PÞÞp4hðxðPÞÞ þ 2 maxflog jaj; log jbjg þ log 16:

ð8Þ

Following the proof of [15, Proposition 4.12], let CN ; CD ; C 0N ; C 0D be integers of minimal height satisfying the Be´zout identities CN N þ CD D ¼ Ca3 p7 ;

C 0N N þ C 0D D ¼ Cq7 ;

ð9Þ

where C :¼ 108a4 72a2 b2 þ 32ab3 4b4 : By a direct computation we obtain the following estimates: jCN jp664 maxfjaj; jbjg5 maxfjpj; jqjg3 ; jCD jp650 maxfjaj; jbjg5 maxfjpj; jqjg3 ; jC 0N jp40 maxfjaj; jbjg2 maxfjpj; jqjg3 ; jC 0D jp38 maxfjaj; jbjg2 maxfjpj; jqjg3 :

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This implies 1314 maxfjaj; jbjg5 maxfjpj; jqjg3 maxfjNj; jDjg ; jpj7 p jCjja3 j

ð10Þ

78 maxfjaj; jbjg2 maxfjpj; jqjg3 maxfjNj; jDjg jqj7 p : jCj

ð11Þ

Now we are going to express these estimates in terms of the height of N=D: Let g be the gcd of N and D: Then (9) shows that g divides Ca3 p7 and Cq7 ; i.e. g divides Ca3 : Let n :¼ N=g and d :¼ D=g: Then we have N ¼ ngpnCa3 ;

D ¼ dgpdCa3 :

Combining these estimates with inequalities (10) and (11) we obtain jpj7 p1314 maxfjaj; jbjg5 maxfjpj; jqjg3 maxfjnj; jdjg; jqj7 p78 maxfjaj; jbjg5 maxfjpj; jqjg3 maxfjnj; jdjg; maxfjpj7 ; jqj7 gp1314 maxfjaj; jbjg5 maxfjpj; jqjg3 maxfjnj; jdjg:

ð12Þ

Since n and d are coprime, hðxð½2PÞÞ ¼ hðN=DÞ ¼ hðn=dÞ ¼ maxflog jnj; log jdjg: Taking logarithms in inequality (12) we obtain 4hðxðPÞÞphðxð½2PÞÞ þ 5 maxflog jaj; log jbjg þ logð1314Þ: Combining this estimate with inequality (8) we deduce the following estimate: jhð½2PÞ 4hðPÞjp5 maxflog jaj; log jbjg þ logð1314Þ:

ð13Þ

Let now PAEðQÞ be a 2-torsion point. Then xðPÞ is a root of the polynomial x3 þ ðb=aÞx2 þ ðb=aÞx þ 1: We conclude that hðxðPÞÞpmaxflog jaj; log jbjg þ 2: This implies that estimate (13) also holds in this case. Finally, combining estimate (13) and Lemma 1 finishes the proof of the lemma. & In order to find to set of Q-rational points of the curve Ct we are going to follow Dem’janenko–Manin’s method [3,7,20]. For this purpose we consider the morphisms f1 ; f2 : Ct -E t defined by
 1 y 2 f1 ðx; yÞ :¼ ðx ; yÞ; f2 ðx; yÞ :¼ ; : x2 x3 The application of Dem’janenko–Manin’s method requires an estimate on the difference hðf1 ðPÞ þ f2 ðPÞÞ 4hðPÞ for any PACt ðQÞ; which is the content of our next result.

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Lemma 4. With notations and assumptions as above, for any point PACt ðQÞ the following inequality holds: jhðf1 ðPÞ þ f2 ðPÞÞ 4hðPÞjp2hðtÞ þ logð62Þ: Proof. Let t :¼ b=a and let P :¼ ðxðPÞ; yðPÞÞ be a Q-rational point of the curve Ct : Suppose first that xðPÞ ¼ 1: Then f1 ðPÞ ¼ f2 ðPÞ and hðPÞ ¼ 0: We conclude that the statement of Lemma 4 holds in this case. Suppose now that xðPÞa 1 holds. Then we have

xðf1 ðPÞ þ f2 ðPÞÞ ¼

2axðPÞ3 þ ðb 3aÞxðPÞ2 2axðPÞ axðPÞ4 þ 2axðPÞ3 þ 2axðPÞ2 þ 2axðPÞ þ a

:

ð14Þ

Let us write xðPÞ ¼ p=q; where p and q are coprime integers. Rewriting identity (14) in terms of p and q we obtain xðf1 ðPÞ þ f2 ðPÞÞ ¼

2ap3 q þ ðb 3aÞp2 q2 2apq3 : ap4 þ 2ap3 q þ 2ap2 q2 þ 2apq3 þ aq4

Let N :¼ 2ap3 q þ ðb 3aÞp2 q2 2apq3 and D :¼ ap4 þ 2ap3 q þ 2ap2 q2 þ 2apq3 þ aq4 : Then xðf1 ðPÞ þ f2 ðPÞÞ ¼ N=D and we have the estimates jNjp ð2jaj þ jb 3aj þ 2jajÞ maxfjpj; jqjg4 p 8 maxfjaj; jbjg maxfjpj; jqjg4 ;

jDjp ðjaj þ 2jaj þ 2jaj þ 2jaj þ jajÞ maxfjpj; jqjg4 p 8 maxfjaj; jbjg maxfjpj; jqjg4 : This implies hðf1 ðPÞ þ f2 ðPÞÞp4hðPÞ þ maxflog jaj; log jbjg þ log 8:

ð15Þ

In order to prove the converse inequality, let CN ; CD ; C 0N ; C 0D be integers of minimal height satisfying the Be´zout identities: CN N þ CD D ¼ Cp7 ;

C 0N N þ C 0D D ¼ Cq7 ;

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where C :¼ 3a3 þ 2a2 b ab2 : By a direct computation we obtain the estimates jCN jp28 maxfjaj; jbjg2 maxfjpj; jqjg3 ; jCD jp34 maxfjaj; jbjg2 maxfjpj; jqjg3 ; jC 0N jp28 maxfjaj; jbjg2 maxfjpj; jqjg3 ; jC 0D jp34 maxfjaj; jbjg2 maxfjpj; jqjg3 : Therefore we have 62 maxfjaj; jbjg2 maxfjpj; jqjg3 maxfjNj; jDjg : maxfjpj7 ; jqj7 gp C Let g be the gcd of N and D: Then g divides Cp7 and Cq7 : Since p and q are coprime, we conclude that g divides C: Let n; d be the integers such that N ¼ ng and D ¼ dg: Then we have maxfjpj7 ; jqj7 gp62 maxfjaj; jbjg2 maxfjpj; jqjg3 maxfjnj; jdjg: Since n and d are coprime we see that hðxðf1 ðPÞ þ f2 ðPÞÞÞ ¼ hðN=DÞ ¼ maxfjnj; jdjg holds. Therefore, taking logarithms in the previous inequality we deduce the following estimate: 4hðPÞphðf1 ðPÞ þ f2 ðPÞÞ þ 2 max logfjaj; jbjg þ logð62Þ: Combining this estimate with (15) finishes the proof of the lemma.

&

Now we are ready to obtain an estimate on the height of the points of Ct ðQÞ:

Theorem 5. Let t be a rational number such that the elliptic curve E t has rank 1 over Q: Then for any point PACt ðQÞ the following estimate holds: 7hðtÞ þ logð81468Þ : hðPÞp 2 Proof. Let f1 ; f2 : Ct -E t be the morphisms f1 ðx; yÞ :¼ ðx2 ; yÞ and f2 ðx; yÞ :¼ ð1=x2 ; y=x3 Þ previously introduced. Let P be a fixed point of Ct ðQÞ: Following the Dem’janenko–Manin’s method, we shall consider the matrix Hˆ :¼ ð/fi ðPÞ; fj ðPÞSÞ1pi; jp2 AC22 ; where /; S denotes the Ne´ron–Tate pairing on E t : Since E t has rank 1, we have that the points f1 ðPÞ; f2 ðPÞAE t ðQÞ are Z-linear dependent. Therefore, from the positive–definiteness of the Ne´ron–Tate pairing on E t ðQÞ=E t ðQÞtors we conclude that the matrix Hˆ is singular. Let us observe that Hˆ can

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be rewritten as 1 ˆ ðPÞ þ f ðPÞÞ hðf ˆ ðPÞÞ hðf 1 2 1 C B ˆ ðPÞÞ C B

hðf 2 C: Hˆ :¼ B C Bˆ ˆ ðPÞÞ A @ hðf1 ðPÞ þ f2 ðPÞÞ hðf 1 ˆ ðPÞÞ ˆ ðPÞÞ

hðf 2hðf 0

ˆ ðPÞÞ 2hðf 1

2

2

Let HAC22 be the following matrix: 0

2hðf1 ðPÞÞ

B B H :¼ B @ hðf1 ðPÞ þ f2 ðPÞÞ hðf1 ðPÞÞ

hðf2 ðPÞÞ

hðf1 ðPÞ þ f2 ðPÞÞ hðf1 ðPÞÞ

hðf2 ðPÞÞ

1 C C C: A

2hðf2 ðPÞÞ

From Lemma 3 we have the estimates: ˆ ðPÞÞjo5hðtÞ þ logð1314Þ; jhðfi ðPÞÞ hðf i 3

ði ¼ 1; 2Þ;

ˆ ðPÞ þ f ðPÞÞjo5hðtÞ þ logð1314Þ: jhðf1 ðPÞ þ f2 ðPÞÞ hðf 1 2 3 We conclude that the entries of the matrix H Hˆ are real numbers of absolute value bounded by 5hðtÞ þ logð1314Þ: From the definition of f1 ; f2 we see that hðf1 ðPÞÞ ¼ hðf2 ðPÞÞ ¼ 2hðPÞ holds. We deduce that H can be expressed as H ¼ K þ 4hðPÞI; where K is the antidiagonal matrix whose non-zero entries are hðf1 ðPÞ þ f2 ðPÞÞ 4hðPÞ and I denotes the ð2  2Þ-identity matrix. Applying Lemma 4 we conclude that the entries of the matrix K are real numbers of absolute value bounded by 2hðtÞ þ logð62Þ: Let L :¼ Hˆ H þ K: Then the entries of L are real numbers of absolute value bounded by 7hðtÞ þ logð81468Þ and the matrix Hˆ can be written as Hˆ ¼ L þ 4hðPÞI: For a given matrix M :¼ ðmi; j Þ1pi; jp2 AC22 ; let us denote by jjMjj the standard N-matrix norm of M: We have jjMjjp2 maxfjmi; j j : 1pi; jp2g: Assuming without loss of generality that hðPÞa0; we see that the matrix ð4hðPÞÞ 1 L þ I ¼ ð4hðPÞÞ 1 Hˆ is singular. This implies jjð4hðPÞÞ 1 LjjX1 (see e.g. [12]). Since the entries of the matrix ð4hðPÞÞ 1 L are real numbers of absolute value bounded by ð4hðPÞÞ 1 ð7hðtÞ þ logð81468ÞÞ we deduce the estimate hðPÞpð7hðtÞ þ logð81468ÞÞ=2: & From Theorem 5 we shall deduce our first uniform upper bound on the number of rational points of the family of curves fCt gtAQ : For this purpose, we need the following technical result:

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Lemma 5. Let G :¼ ð0; 1ÞAE t ðQÞ: Then the following estimate holds: ˆ hðGÞXðhðtÞ

17:94Þ=12: Proof. Let t :¼ b=a; with a; bAZ and gcdða; bÞ ¼ 1: The x-coordinate of the point ½2G is given by xð½2GÞ ¼ ð 4ab þ b2 Þ=4a2 : Let N :¼ 4ab þ b2 ; D :¼ 4a2 ; and let CN ; CD ; C 0N ; C 0D be integers of minimal height satisfying the Be´zout identities CN N þ CD D ¼ 4a2 ;

C 0N N þ C 0D D ¼ b3 :

By a direct computation we obtain the estimates 4jaj2 pjDj;

jbj3 pð5 þ 4Þ maxfjaj; jbjg maxfjNj; jDjg:

This implies that maxfjaj; jbjg2 p 9 maxfjNj; jDjg holds. Therefore, we have 2 maxflog jaj; log jbjgplogð9Þ þ maxflog jDj; log jNjg: Let g be the gcd of N and D and let n :¼ N=g; d :¼ D=g: Then g divides 4a2 and b3 ; and hence divides 4: This implies 2 maxflog jaj; log jbjgplogð36Þ þ maxflog jdj; log jnjg: Since n and d are coprime, the above inequality may be rewritten as hð½2GÞX2hðtÞ logð36Þ: From Lemma 3 we have the estimate 5 logð1314Þ ˆ : hð½2GÞXhð½2GÞ

hðtÞ

3 3 ˆ ˆ Combining the last two estimates, the identity 4hðGÞ ¼ hð½2GÞ and the inequality logð61305984Þo17:94; we easily deduce the statement of the lemma. & Let PCQ be the set of values t for which the elliptic curve E t has rank 1 over Q and G :¼ ð0; 1Þ is a generator of the free part of the group E t ðQÞ: In Section 5 we discuss in a statistical sense how many rationals belong to P: We have the following result concerning the family of curves fCt gtAP : Corollary 1. There exists NAN such that for any tAP we have #Ct ðQÞpN: Proof. Let tAP; let G :¼ ð0; 1ÞAE t and let us fix a point PACt ðQÞ: Let f1 : Ct -E t be the morphism defined by f1 ðx; yÞ :¼ ðx2 ; yÞ: Then there exists nAN and T AE t ðQÞtors ˆ ˆ ðPÞÞ ¼ n2 hðGÞ: such that f1 ðPÞ ¼ ½nG þ T holds. Therefore, we have hðf 1

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ˆ ðPÞÞ: On one hand, estimate (13) implies Now we estimate the quantity hðf 1 ˆ hðf1 ðPÞÞ hðf1 ðPÞÞp5hðtÞ=3 þ logð1314Þ=3: On the other hand, Theorem 5 yields the estimate hðf1 ðPÞÞ ¼ 2hðPÞp7hðtÞ þ logð81468Þ: Putting together these estimates we obtain ˆ ðPÞÞp26hðtÞ þ 13:71: hðf 1 3

ð16Þ

1 ˆ Let tAP satisfy the condition hðtÞ418:94: By Lemma 5 we have hðGÞ p12ðhðtÞ

1 17:94Þ ; from which we deduce

n2 p104

hðtÞ þ 1:59 : hðtÞ 17:94

ð17Þ

Since the right-hand side of the last estimate is a bounded quantity for any tAQ with hðtÞ418:94; we conclude that the cardinality of the set Ct ðQÞ is uniformly bounded in the set of values tAP with hðtÞ418:94: On the other hand, the set of values tAQ such that hðtÞp18:94 holds is finite. Hence the cardinality of the set Ct ðQÞ is uniformly bounded in the set of values tAQ with hðtÞp18:94: This concludes the proof of the corollary. & Remark 1. From (17) we easily conclude that for all but finitely many tAP the estimate np10 holds. 4.2. The structure of Ct ðQÞ In this section we prove Theorem 3, which determines the arithmetic structure of the curve Ct for all but finitely many values tAP; where P is the set of rational numbers t for which the elliptic curve E t has rank 1 and (0,1) is a generator of the free part of the group E t ðQÞ: 4.2.1. The torsion subgroup of E t ðQÞ In order to determine the group Ct ðQÞ we first describe the torsion group E t ðQÞtors : This is the subject of the following proposition. Proposition 2. For all but finitely many tAQ the following assertions hold: (i) if there exists uAQ\f0; 1; 1g such that t ¼ ðu2 u þ 1Þ=u holds, then 
 1 E t ðQÞtors ¼ OE t ; ð 1; 0Þ; ðu; 0Þ; ; 0 ; u all points having order 2. (ii) Otherwise, we have E t ðQÞtors :¼ fOE t ; ð 1; 0Þg:

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Proof. Mazur’s Theorem [21] asserts that the torsion subgroup of E t ðQÞ is isomorphic to one of following groups: * *

Z=mZ; with 1pmp10 or m ¼ 12; Z=2Z  Z=2mZ; with 1pmp4:

The point P0 :¼ ð 1; 0ÞAE t ðQÞ is a torsion point of order 2. This restricts the choices for the torsion subgroup of E t ðQÞ to Z=mZ with mAf2; 4; 6; 8; 10; 12g and Z=2Z  Z=mZ with mAf1; 2; 3; 4g: The following lemma restricts further the possible torsion subgroups. Lemma 6. For all but finitely many tAQ the torsion subgroup E t ðQÞtors of the group E t ðQÞ is isomorphic to Z=2Z or Z=2Z  Z=2Z: Proof. Suppose that the torsion group E t ðQÞtors is not isomorphic to one of the groups Z=2Z or Z=2Z  Z=2Z: Then, the above remarks show that E t ðQÞtors has necessarily elements of order 3, 4 or 5. Let i be any of the values 3, 4 or 5. We claim that the set of values tAQ such that there exists a torsion point of E t ðQÞ of order i is finite. We sketch the strategy of the proof of the general case and detail the computations in the case i ¼ 3: Let P :¼ ðxðPÞ; yðPÞÞ be a point in E t ðQÞtors : Then P is an i-torsion point if and only if the i-torsion polynomial pi ðt; xÞ of the elliptic curve E t ðQÞ vanishes in xðPÞ: A direct computation shows that for any iAf3; 4; 5g the equation pi ðt; xÞ ¼ 0 defines ðiÞ ðiÞ genus-0 curve CðiÞ : Let x ¼ v1 ðuÞ; t :¼ v2 ðuÞ be a parametrization of the curve CðiÞ ; ðiÞ

ðiÞ

where v1 ; v2 are suitable rational functions of QðuÞ: Replacing this parametrization in the equation y2 ¼ x3 þ tx2 þ tx þ 1 of the elliptic curve E t we obtain a plane curve y2 ¼ vðiÞ ðuÞ which is an elliptic curve of rank 0. This implies that there exists a finite set of Q-rational points ðu; yÞ satisfying the equation y2 ¼ vðiÞ ðuÞ and thus a finite set of Q-rational points ðt; xÞ satisfying the equation pi ðt; xÞ ¼ 0: Therefore the set of points ðxðPÞ; yðPÞ; tÞAQ3 such that P :¼ ðxðPÞ; yðPÞÞ is a torsion point of order i of the curve E t is finite. We conclude that set of values tAQ for which the curve E t has torsion points of order i is finite. Now we detail the computations for the case i ¼ 3: In this case the 3-division polynomial is p3 ðt; xÞ :¼ 3x4 þ 4tx3 þ 6tx2 þ 12x t2 þ 4t: The equation p3 ðx; tÞ ¼ 0 defines a plane curve of genus 0 which can be parametrized as follows: x¼

ð 4 þ 3uÞðu þ 4Þ ; 16u

t¼

ð 4 þ 3uÞð3u3 12u2 þ 144u 64Þ : 64u3

Replacing this parametrization in the equation y2 ¼ x3 þ tx2 þ tx þ 1 defining the elliptic curve E t we obtain the plane curve y2 ¼

ðu 4Þ2 ð3u2 þ 24u 16Þ3 : 16384u5

ð18Þ

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Making the change of variables y ¼ ðu 4Þð3u2 þ 24u 16ÞY =128u3 we see that the non-zero rational solutions of (18) are in bijection with the rational solutions of the curve Y 2 ¼ 3u3 þ 24u2 16u: Taking into account that this is an elliptic of rank 0 over Q finishes the proof of our assertion in the case i ¼ 3: & Now we can complete the proof of Proposition 2. By Lemma 6 for all but a finite set of values tAQ the torsion group E t ðQÞtors is isomorphic to one of the groups Z=2Z and Z=2Z  Z=2Z: Let us fix a value tAQ such that the group E t ðQÞtors is isomorphic to the group Z=2Z  Z=2Z: Then E t ðQÞtors has three distinct elements of order 2, whose x-coordinates are three distinct rational roots of the polynomial p2;t ðxÞ :¼ x3 þ tx2 þ tx þ 1 ¼ ðx þ 1Þðx2 þ tx x þ 1Þ: In such a case, there exists a root uAQ\f0; 1; 1g of the polynomial p2;t and hence t ¼ ðu2 u þ 1Þ=u holds (observe that the values u ¼ 71 make the curve E t singular). We easily conclude that the torsion subgroup of E t ðQÞ is 
 1 E t ðQÞtors ¼ OE t ; ð 1; 0Þ; ðu; 0Þ; ; 0 : u On the other hand, if the group E t ðQÞtors is isomorphic to Z=2Z; taking into account that ð 1; 0Þ is a non-trivial torsion point of E t ðQÞ we conclude that E t ðQÞtors ¼ fOE t ; ð 1; 0Þg holds. This completes the proof of Proposition 2. &

4.2.2. The set Ct ðQÞ Now we are able to prove Theorem 3, which determines the set of Q-rational points of the curve Ct for all but finitely many values tAP: Theorem 3. For all but finitely many values tAP the following assertions hold: (i) if there exists vAQ such that t ¼ ðv4 v2 þ 1Þ=v2 holds, then 

 1 1 Ct ðQÞ ¼ ð0; 1Þ; ð0; 1Þ; ðv; 0Þ; ð v; 0Þ; ; 0 ; ; 0 : v v (ii) Otherwise, we have Ct ðQÞ ¼ fð0; 1Þ; ð0; 1Þg: Proof. Let tAQ and let as before f1 ; f2 : Ct -E t denote the morphisms defined by f1 ðx; yÞ :¼ ðx2 ; yÞ and f2 ðx; yÞ :¼ ð1=x2 ; y=x3 Þ: Observe that for any point P ¼ ðxðPÞ; yðPÞÞ of Ct ðQÞ we have f1 ðPÞAE t ðQÞ and f2 ðPÞAE t ðQÞ: Corollary 1 and

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Remark 1 show that for all but a finite set of values tAP the points f1 ðPÞ and f2 ðPÞ can be expressed as f1 ðPÞ ¼ ½n1 ð0; 1Þ þ T 1 and f2 ðPÞ ¼ ½n2 ð0; 1Þ þ T 2 ; with jn1 j; jn2 jp10 and T 1 ; T 2 AE t ðQÞtors : Let us fix for the moment an integer n and a torsion point T :¼ ðt1 ; t2 Þ of E t : Then the x-coordinate of the point ½nð0; 1Þ þ T AE t ðQÞ can be expressed as a rational function in the value t; which we denote by Fn;T ðtÞ: We shall see that for any point PACt ðQÞ the definition of the morphisms f1 ; f2 imply that there exist T 1 ; T 2 AE t ðQÞtors such that the condition Fn1 ;T 1 ðtÞFn2 ;T 2 ðtÞ ¼ 1 is satisfied. The existence of this algebraic condition on the value t is a key point of the proof of Theorem 3. Proof of Theorem 3. (i) Let tAP and let us suppose that there exists vAQ such that t ¼ ðv4 v2 þ 1Þ=v2 : Letting u :¼ v2 we see that there exists uAQ\f0; 1; 1g for which t ¼ ðu2 u þ 1Þ=u holds. Then Proposition 2(i) shows that the torsion  subgroup of E t ðQÞ is given by E t ðQÞtors ¼ OE t ; ð 1; 0Þ; ðu; 0Þ; 1u; 0 ¼: fT 1 ; T 2 ; T 3 ; T 4 g; all points having order 2. Then any point T AE t ðQÞtors has order at most 2 and we have that for any nAZ the x-coordinates of the points ½nð0; 1Þ þ T and ½ nð0; 1Þ þ T agree. Therefore, in order to determine which are the possible xcoordinates of the image of a point PACt ðQÞ we may assume without loss of generality that nX0 holds. For 1pip4 and 0pnp10; let Fn;i ðuÞ denote the rational function which represents the x-coordinate of the point ½nð0; 1Þ þ T i : Let P :¼ ðxðPÞ; yðPÞÞ be a point of Ct ðQÞ: Then Proposition 2(i) and Remark 1 show that for all but finitely many values tAP we have that xðPÞ and u satisfy the condition: xðPÞ2 ¼ Fn1 ; j1 ðuÞ;

1 xðPÞ2

¼ Fn2 ; j2 ðuÞ;

ð19Þ

with 0pn1 ; n2 p10 and j1 ; j2 Af1; 2; 3; 4g: Let us observe that the cases n1 ¼ 0; j1 ¼ 1 and n2 ¼ 0; j2 ¼ 1 cannot arise because the point OE t ¼ ½0ð0; 1Þ does not belong to the affine part of the curve E t : On the other hand, the cases n1 ¼ j1 ¼ 1 and n2 ¼ j2 ¼ 1 yield the point ð0; 1Þ ¼ ½1ð0; 1Þ; which is the image of the points ð0; 71ÞACt ðQÞ: Finally, the cases n1 ¼ 0; j1 ¼ 2 and n2 ¼ 0; j2 ¼ 2 cannot arise because the x-coordinate of the point ½0ð0; 1Þ þ ð 1; 0Þ ¼ ð 1; 0Þ is not a square in Q: In all the remaining cases (19) shows that the equation Fn1 ; j1 ðuÞFn2 ; j2 ðuÞ ¼ 1

ð20Þ

holds. A direct computation shows that this identity is satisfied for all the values uAQ if and only if n1 ¼ n2 ¼ 0 and j1 ¼ 3; j2 ¼ 4 or j1 ¼ 4; j2 ¼ 3 hold. In all the other cases Fn1 ; j1 ðuÞFn2 ; j2 ðuÞ 1 is a non-zero rational function which vanishes in a finite set values uAQ: Since there are only a finite set of possible choices for the integers n1 ; n2 ; j1 ; j2 ; we conclude that for all but finite many values uAQ identity (20) will not be satisfied unless n1 ¼ n2 ¼ 0 and j1 ¼ 3; j2 ¼ 4 or j1 ¼ 4; j2 ¼ 3 hold. In this latter case the conditions x2 ¼ F0;3 ðuÞ ¼ u or x2 ¼ F0;4 ðuÞ ¼ u are

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satisfied if and only if u is a square in Q; which holds true since by assumption u ¼ v2 : Taking into account that the fibre of the set fðu; 0Þ; ð1=u; 0Þg under the morphisms f1 ; f2 is the set fð7v; 0Þ; ð71=v; 0Þg we easily conclude the statement of Theorem 3(i). (ii) Now we have that there does not exist vAQ such that t ¼ ðv4 v2 þ 1Þ=v2 : If there exists uAQ for which t ¼ ðu2 u þ 1Þ=u holds, the arguments of the proof of Theorem 3(i) show that Ct ðQÞ ¼ fð0; 1Þ; ð0; 1Þg holds. Therefore, we may assume without loss of generality that there does not exist uAQ such that t ¼ ðu2 u þ 1Þ=u holds. Then Proposition 2(ii) shows that E t ðQÞtors ¼ fOE t ; ð 1; 0Þg holds. Let us fix nAZ: Then there exist rational functions Fn;1 ; Fn;2 AQðtÞ which represent the x-coordinate of the points ½nð0; 1Þ and ½nð0; 1Þ þ ð 1; 0Þ; respectively. Arguing as before we conclude that without loss of generality we may assume that nX0 holds. Let P :¼ ðxðPÞ; yðPÞÞ be a point in Ct ðQÞ: From Remark 1 we deduce that xðPÞ and t satisfy the relation x2 ðPÞ ¼ Fn1 ; j1 ðtÞ;

1 ¼ Fn2 ; j2 ðtÞ x2 ðPÞ

ð21Þ

with 0pn1 ; n2 p10 and j1 ; j2 Af1; 2g: We observe that the cases n1 ¼ 0; j1 ¼ 1 and n2 ¼ 0; j2 ¼ 1 do not yield points of Ct ðQÞ; because the point ½0ð0; 1Þ does not belong to the affine part of the elliptic curve E t : On the other hand, the cases n1 ¼ 0; j1 ¼ 2 and n2 ¼ 0; j2 ¼ 2 do not yield points of Ct ðQÞ; because the xcoordinate of the point ½0ð0; 1Þ þ ð 1; 0Þ ¼ ð 1; 0Þ is not a square in Q: Finally, in the case n1 ¼ j1 ¼ 1 we have the point ð0; 1ÞAE t ðQÞ; whose f1 -fibre is the set fð0; 1Þ; ð0; 1Þg for any tAQ: In all the remaining cases (21) implies Fn1 ; j1 ðtÞFn2 ; j2 ðtÞ ¼ 1: Furthermore, in all these cases Fn1 ; j1 ðtÞFn2 ; j2 ðtÞ 1 is a non-zero element of QðtÞ; thus vanishing in a finite set of values tAQ: Since there are only a finite set of admissible choices for the integers n1 ; n2 ; j1 ; j2 we conclude that for all but a finite set of values tAQ the identity Ct ðQÞ ¼ fð0; 1Þ; ð0; 1Þg holds. This concludes the proof of Theorem 3(ii). &

5. Experimental and conjectural results Theorem 3 asserts that the cardinality of the set Ct ðQÞ is uniformly bounded in the set of values tAQ satisfying the following conditions: 1. The rank of the abelian group E t ðQÞ is 1. 2. (0,1) is a generator of the free part E t ðQÞ: The purpose of this section is twofold. On one hand, we are going to discuss the ‘‘strength’’ of conditions 1 and 2 from a experimental point of view. On the other hand, we are going to show that under the assumption of the validity of Conjecture B condition 2 is not necessary.

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5.1. Rank considerations Since Theorem 2 shows that conditions 1 and 2 are satisfied by the elliptic curve E defined over QðTÞ; one might expect these conditions to frequently happen over Q i.e. for the specialized Q-definable curves E t : Unfortunately, this needs not be true. e defined over QðTÞ Indeed, Cassels and Schinzel [5] exhibit a rank-0 elliptic curve E with the following property: assuming Selmer’s conjecture [24], for any tAQ the et has rank at least 1. specialized curve E The general question of characterizing the behaviour of the rank of an elliptic curve defined over QðTÞ under specializations is a difficult problem (see e.g. [30]). Nevertheless, there is some numerical experience, as that of Fermigier [9] who studies et ; with tAZ; coming from 93 QðTÞ-definable elliptic curves E e 66,918 elliptic curves E having ranks between 0 and 4 over QðTÞ: Fermigier shows that, with a surprising amount of uniformity, the following identity holds: eðQðTÞÞ þ N; et ðQÞ ¼ rank E rank E where N ¼ 0 with probability 32%; N ¼ 1 with probability 48%; N ¼ 2 with probability 18%; N ¼ 3 with probability 2%: We computed the rank of 284,051 elliptic curves E t with hðtÞplogð530Þ: We obtain the following results: rank E t ðQÞ ¼ rank EðQðTÞÞ þ N; where N ¼ 0 with probability 32:7%; N ¼ 1 with probability 49:9%; N ¼ 2 with probability 15:9%; N ¼ 3 with probability 1:5%: These figures suggest that condition 1 might hold with a probability of success of approximately 1=3: This seems to contradict the Katz–Sarnak conjectures [13,14], which probably imply that N ¼ 0 and 1 should each occur with probability 50%; and NX2 should occur with probability 0. However, there is some reason to believe that there is a ‘‘small number phenomenon’’ at work here, probably due to the existence

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of small degree subfamilies of higher rank. Indeed, as a referee has pointed out to us, the elliptic surface defined over QðUÞ of equation y2 ¼ x3 þ ð2U 2 1Þx2 þ ð2U 2

1Þx þ 1 has rank 2. Since our experimental results are concerned with values of tAQ with hðtÞplogð530Þ; even this one subfamily will yield a considerable number of curves of rank 2, although it contributes nothing to the density. We refer to [36] for further discussion on the average rank of a family of elliptic curves. 5.2. Divisibility considerations If the point (0,1) is a generator of the free part of the group EðQðTÞÞ; the same statement does not necessarily hold in a specialized curve E t : even if the elliptic curve E t has rank 1 over Q; the point (0,1) could be a multiple of a generator of the free part of E t ðQÞ: e be a elliptic curve defined over This problem can be put into a general setting: let E et is an elliptic curve QðTÞ; then for all but finitely many tAQ; the specialized curve E defined over QðTÞ and we may consider the specialization homomorphism st : eðQðTÞÞ/E et ðQÞ: E et ðQÞ for values tAN; In [30], Silverman asks whether the image of st is divisible in E et ðQÞ such that ½nPAst ðE eðQðTÞÞÞ for some integer i.e. whether there are points PAE eðQðTÞÞÞ for tAN: Assuming that Conjecture B is true, Theorems 2 nX2 and Pest ðE and 3 of [30] give the following result. Theorem 6. Let notations and assumptions as above. Assume that Conjecture B is true, e has non-constant j-invariant. Then the following and suppose that the elliptic curve E assertions hold: (i) There exists an absolute constant C40 with the following property: for any tAQ eðQðTÞÞÞ#Q holds, there exists 0pnoC such and any PAE t ðQÞ for which PAst ðE eðQðTÞÞÞ holds. that ½nPAst ðE eðQðTÞÞÞ is indivisible in E et ðQÞ has density 1. (ii) The set of values tAQ for which st ðE Proof (Sketch). We shall follow mutatis mutandis the proof of [30, Theorem 2], indicating briefly the necessary changes. We observe that Facts 2 and 6 of that proof also hold for values tAQ (see e.g. [35, Chapter III, Corollary 11.3.1; 31, Chapter VIII, Theorem 5.6]). Therefore, taking into account that Conjecture B implies Fact 3 of the proof [30, Theorem 2] for any tAQ; we easily conclude that (i) holds. In order to prove (ii), we observe that the proof of the Neron’s specialization Theorem of [25, Chapter 11.1] shows that for any nAN; the set On of values tAQ for which the specialization morphism EðQðTÞÞ=nEðQðTÞÞ-E t ðQÞ=nE t ðQÞ is injective is a thin set. Therefore, the estimates of [25, Section 9.7, Theorem, p. 133] imply that On has density 1 in Q: Combining this with the proof of [30, Theorem 2], we see that (ii) holds. &

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Applying Theorem 6 to the elliptic curve E of equation y2 ¼ x3 þ Tx2 þ Tx þ 1 we obtain the following result: Corollary 2. Assume that Conjecture B is true. Let Q denote the set of values tAQ such that the abelian group E t ðQÞ has rank 1 and let R denote the (density 1) set of values tAQ for which st ðEðQðTÞÞÞ is indivisible in E t ðQÞ: (i) For any tAR-Q; the point ð0; 1Þ generates the free part of E t ðQÞ: eAN such that the following property holds: for any tAQ; if Gt is a (ii) There exists C e such that ð0; 1Þ

generator of the free part of E t ðQÞ then there exists npC ½nGt AE t ðQÞtors holds. Proof. Let st : EðQðTÞÞ-E t ðQÞ be the specialization homomorphism of the elliptic curve E: Silverman [29] shows that for all but finitely many values tAQ the homomorphism st is injective. This implies that for all but finitely many values tAQ the subgroup of E t ðQÞ generated by the point ð0; 1Þ is a torsion free subgroup of rank 1. Let tAR-Q and let Gt be a generator of the free part of the group E t ðQÞ: Then there exist mAZ and T AE t ðQÞtors such that ð0; 1Þ ¼ ½mGt þ T holds. Therefore, multiplying this identity by n :¼ 3  5  7  8  11 we conclude that ½nð0; 1Þ ¼ ½nmGt holds. Since ½nmGt ¼ ½nð0; 1ÞAst ðEðQðTÞÞÞ; by the indivisibility of st ðEðQðTÞÞÞ we see that Gt Ast ðEðQðTÞÞÞ holds. Let GAEðQðTÞÞ be such that st ðGÞ ¼ Gt holds. By Proposition 1 we have G ¼ ½sð0; 1Þ þ ½s0 ð 1; 0Þ with sAZ and s0 Af0; 1g: Then we have Gt ¼ ½sst ð0; 1Þ þ ½s0 st ð 1; 0Þ ¼ ½sð0; 1Þ þ ½s0 ð 1; 0Þ: Multiplying this identity by m we have ð0; 1Þ

T ¼ ½mGt ¼ ½msð0; 1Þ þ ½ms0 ð 1; 0Þ: We conclude that the point ð1 msÞð0; 1Þ is a torsion point of E t ðQÞ; which implies ms ¼ 1: From this we easily deduce that the point ð0; 1Þ generates the free part of the group E t ðQÞ: This shows assertion (i). For the second assertion, arguing as above we have that there exists mAZ\f0g and T AE t ðQÞtors such that ½mGt þ T ¼ ð0; 1Þ holds. Then we have ½mnGt Ast ðEðQðTÞÞÞ; where n :¼ 3  4  5  7  11: If Gt Ast ðEðQðTÞÞÞ and GAEðQðTÞÞ satisfies st ðGÞ ¼ Gt ; then there exists s; s0 AZ such that Gt ¼ ½sð0; 1Þ þ ½s0 ð 1; 0Þ holds. Arguing as above we conclude that ms ¼ 1; which implies ð0; 1Þ ½mGt AE t ðQÞtors with jmjp1: Suppose now that Gt est ðEðQðTÞÞÞ holds. Then Theorem 6(i) shows that mnpC 0 holds, where C 0 is the constant of the statement of Theorem 6(i) for the curve E: Thus ð0; 1Þ ½mGt AE t ðQÞtors with jmjpC 0 =n: This concludes the proof of assertion (ii). & We experimentally analyzed the density of the set R-Q of values tAQ for which the rank of E t ðQÞ is 1 and the point ð0; 1Þ generates the free part of the group E t ðQÞ: For this purpose we tested 28469 elliptic curves E t of rank 1 with hðtÞplogð280Þ: We found that the point G :¼ ð0; 1ÞAE t ðQÞ is a generator of the free part of E t ðQÞ in 99.4% of these curves.

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From Corollary 2 we deduce the following result, which shows that if Conjecture B is true then the uniform upper bound of Corollary 1 holds for any tAQ; even in the case that the point ð0; 1ÞAE t ðQÞ does not generate the free part of the group E t ðQÞ:

Theorem 4. Assuming that Conjecture B is true, for any tAQ the cardinality of the set Ct ðQÞ is uniformly bounded. Proof. Let Gt be a generator of the free part of E t ðQÞ: Then Corollary 2(ii) shows that there exists npC such that ð0; 1Þ ½nGt AE t ðQÞtors holds, where C is the ˆ t Þ: Moreover, from the ˆ 1ÞpC 2 hðG constant of Corollary 2(ii). Then we have hð0; ˆ 1Þ 1 p12ðhðtÞ

proof of Corollary 1 we see that if hðtÞ418:94 holds then hð0; 17:94Þ 1 holds. This implies the estimate 12C 2 : p ˆ t Þ hðtÞ 17:94 hðG 1

ð22Þ

Let P be a point of Ct ðQÞ: Then there exist nAN and T AE t ðQÞtors such that f1 ðPÞ ¼ ˆ t Þ: On the other hand, from the ˆ ðPÞÞ ¼ n2 hðG ½nGt þ T holds. Hence we have hðf 1 proof of Corollary 1 we deduce the estimate ˆ ðPÞÞp26 hðtÞ þ 13:71: hðf 1 3

ð23Þ

Let tAQ satisfy the condition hðtÞ418:94: Then estimates (22) and (23) imply n2 p104C 2

hðtÞ þ 1:59 : hðtÞ 17:94

Since the right-hand side of the last estimate is a bounded quantity for any hðtÞ418:94; we conclude that the cardinality of the set Ct ðQÞ can be uniformly bounded for any hðtÞ418:94 such that the rank of the group E t ðQÞ is 1. On the other hand, the set of values tAQ with hðtÞp18:94 is finite and hence the cardinality of the set Ct ðQÞ can be uniformly bounded for such values of t: This concludes the proof of the theorem. &

Acknowledgments The authors thank Joseph Silverman, Noam Elkies and an anonymous referee for their corrections and suggestions that greatly improved the contents and presentation of this paper.

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