Uniform bounds on the number of rational points of a family of curves of genus 2^{∗}

L. Kulesz, G. Matera

Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento

Campus Universitario, José M. Gutiérrez 1150 (1613) Los Polvorines

      Buenos Aires, Argentina, {lkulesz,gmatera}@ungs.edu.ar

É. Schost

Laboratoire GAGE, École polytechnique,

F-91128 Palaiseau Cedex, France, Eric.Schost@polytechnique.fr

Abstract

We exhibit a genus–2 curve C defined over ℚ(T) which admits two independent morphisms to a rank–1 elliptic curve defined over ℚ(T). We describe completely the set of ℚ(T)–rational points of the curve C and obtain a uniform bound for the number of ℚ–rational points of a rational specialization C_{t} of the curve C for a certain (possibly infinite) set of values t ∈ ℚ. Furthermore, for this set of values t ∈ ℚ we describe completely the set of ℚ–rational points of the curve C_{t}. Finally we show how these results can be strengthened assuming a height conjecture of S. Lang.

1 Introduction

In 1983, G. Faltings proved Mordell’s Conjecture, which asserts that for any number field K, the set C(K) of K–rational points of a curve C defined over K of genus at least 2 is finite (see [Fal83]). 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 → ℙ^{1}(ℚ) of curves of (fixed) genus ≥ 2. This question is strongly related to the following conjecture of S. Lang [Lan86]:

Conjecture A Let V be a variety of general type defined over a number field K. Then the set V (K) of K-rational points of V is contained in a subvariety of V of codimension at least 1.

∗Research was partially supported by the following Argentinian and French grants : UBA-CyT X198, PIP CONICET 4571, ECOS A99E06, UNGS 30/3003. Some of the results presented here were first announced at the Workshop Argentino de Informática Teórica, WAIT’01, held in September 2001 (see [GKMS01]).

As an attempt to understand Conjecture A, L. Caporaso, J. Harris and B. Mazur showed the following consequence of this conjecture in the case of algebraic curves (see [CHM95], [CHM97]):

Theorem 1 If Conjecture A is true, then for any number field K and any integer g ≥ 2 there exists an integer B(K, g) such that any non–singular curve defined over K of genus g has at most B(K, g) 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 ≥ 2, were obtained in [Sil87], [Sil93], [Kul99], [Sto01]. These articles consider families of twists of certain fixed curves of genus ≥ 2 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 {C_{t}}_{t∈ℚ} of equation

y^{2} = x^{6} + tx^{4} + tx^{2} + 1.

By means of a direct computation of the invariants of the curve C_{t} we see that for all but finitely many pairs (t, u) ∈ ℚ^{2} with t ≠ u the curves C_{t} and C_{u} are isomorphic over ℂ if and only if u = \frac{15−t}{1+t} holds. Furthermore, this isomorphism is ℚ–definable if and only if 2 + 2t is a square in ℚ. This implies that the family of curves {C_{t}}_{t∈ℚ} contains infinitely many non–ℚ–isomorphic curves.

Let us observe that the family of curves \{C_{t}\}_{t∈\overline{ℚ}} may be intrinsically defined in the following terms: it is (up to \overline{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. [CF96] we see that any \overline{ℚ}–definable genus–2 curve with a degree–2 morphism to an elliptic curve is isomorphic to a curve C_{α,β} of equation y^{2} = x^{6} + αx^{4} + βx^{2} + 1 for suitable α, β ∈ \overline{ℚ}. This implies that the curve C_{α,β} admits two independent degree–2 morphisms to the elliptic curves of equations y^{2} = x^{3} +αx^{2} +βx+1 and y^{2} = x^{3} +βx^{2} +αx+1. Let λ ∈ \overline{ℚ} be such that λ^{2} + λ + 1 = 0. Then the above elliptic curves have the same j–invariant if and only if one of the following conditions hold: (i) β = α; (ii) β = −α − 3; (iii) β = λα or β = −(λ + 1)α; (iv) β = −λα + 3(λ + 1) or β = (λ + 1)α − 3λ.

A direct computation shows that the unidimensional family of curves \{C_{α,β}\}_{α∈\overline{ℚ}} corresponding to the cases (iii) and (iv) is \overline{ℚ}–isomorphic to one of the families corresponding to the cases (i) and (ii). On the other hand, the family of curves corresponding to the case (ii) is mapped into the families of elliptic curves \{E_{α,1}\}_{α∈\overline{ℚ}}, \{E_{α,2}\}_{α∈\overline{ℚ}} of equation y^{2} = x^{3} + αx^{2} + αx + 1 and y^{2} = x^{3} + αx^{2} − (α + 3)x + 1 respectively. Since E_{α,2} does not have any 2– torsion point defined over \overline{ℚ}(α) we conclude that the family \{C_{t}\}_{t∈\overline{ℚ}}, 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 ℚ(T) and \overline{ℚ}(T) denote the field of rational functions in the variable T with coefficients in ℚ and \overline{ℚ} respectively and let \overline{ℚ(T)} denote the algebraic closure of ℚ(T). First we analyze the arithmetic behaviour of the plane curve C defined over ℚ(T) of equation y^{2} = x^{6} + T x^{4} + T x^{2}+1. Our methods rely on the observation that the (independent) morphisms ϕ_{1}, ϕ_{2} defined by

ϕ_{1}(x, y) := (x^{2}, y), ϕ_{2}(x, y) := (\frac{1}{x^{2}} , \frac{y}{x^{3}} ) ,

map the curve C into the elliptic curve E defined over ℚ(T) of equation y^{2} = x^{3} + T x^{2} + T x + 1. We first characterize the structure of the group of ℚ(T)– rational points of E applying Shioda’s theory of Mordell–Weil lattices. Then, using a variant of Dem’janenko–Manin’s method [Dem68, Man69] to find the set of rational points of a given plane curve, we obtain the following result:

Theorem 2 C(ℚ(T)) = {(0, 1), (0, −1)}.

Then for a given value t ∈ ℚ we analyze the arithmetic behaviour of the curve C_{t} using Dem’janenko–Manin’s method. For this purpose, we observe that the restriction ϕ_{1}|_{C∩\overline{ℚ}^{2}}, ϕ_{2}|_{C∩\overline{ℚ}^{2}} of the morphisms ϕ_{1}, ϕ_{2} defined above map the curve C_{t} into the elliptic curve E_{t} defined over ℚ of equation

y^{2} = x^{3} + tx^{2} + tx + 1.

For any value t ∈ ℚ such that the abelian group E_{t}(ℚ) of ℚ–rational points of the elliptic curve E_{t} has rank 1 and its free part is generated by the point (0, 1), we determine the set C_{t}(ℚ) of ℚ–rational points of the curve C_{t}. We prove the following result:

Theorem 3 Let P ⊂ ℚ denote the set of all t ∈ ℚ such that the abelian group E_{t}(ℚ) has rank 1 and its free part is generated by the point (0, 1). Then the following statements hold for all but finitely many t ∈ P:

(i) If there exists v ∈ ℚ such that t = −(v^{4} − v^{2} + 1)/v^{2} holds, then

C_{t}(ℚ) = \{(0, 1), (0, −1), (v, 0), (−v, 0), (\frac{1}{v} , 0), ( − \frac{1}{v} , 0)\} .

(ii) Otherwise, we have

C_{t}(ℚ) = {(0, 1), (0, −1)}.

Let h and ĥ denote the naive (logarithmic) height on ℚ and the canonical height on a given elliptic curve Ẽ defined over ℚ respectively (see the next section for precise definitions). Then the statement of Theorem 3 can be significantly improved for values t ∈ ℕ assuming that the following conjecture of S. Lang holds [Lan78]:

Conjecture B There exists a universal constant c > 0 such that for any elliptic curve Ẽ defined over ℚ of discriminant Δ and any nontorsion point P ∈ Ẽ(ℚ), the estimate ĥ(P) > c · h(Δ) holds.

Let us observe that Conjecture B has been proved for elliptic curves with integral j–invariant [Sil94]. Furthermore, [HS88] 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 (0, 1) is a generator of the free part of the group E_{t}(ℚ) is not essential for t ∈ ℕ:

Theorem 4 If Conjecture B is true there exists a universal constant C > 0 with the following property: for any t ∈ ℕ such that the abelian group E_{t}(ℚ) has rank 1, the cardinality of the set C_{t}(ℚ) 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 t ∈ ℚ:

1. The rank of the abelian group E_{t}(ℚ) is 1.

2. (0,1) is a generator of the free part of E_{t}(ℚ).

In Section 5 we discuss how restrictive these conditions on the parameter t ∈ ℚ are. Theorem 4 shows that our uniform upper bound on the cardinality of the set C_{t}(ℚ) 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 t ∈ ℚ for which E_{t}(ℚ) has rank 1. Our experimental results seem to show that the set of values t ∈ Q for which (0,1) is a generator of the free part of E_{t}(ℚ) 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 J. Cremona’s software mwrank [Cre] and took about two months of CPU time on a 1Ghz PC. All the other symbolic computations were done using the Magma computer algebra system [Mag]. All software and hardware resources were provided by the French computation center MEDICIS [MED].

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 [Kna92],

[Sil86] and [Sil94].

Let K denote any of the fields ℚ or ℚ(T) and let O_{K} denote its ring of integers i.e. ℤ or ℚ[T] respectively. For x = x_{1}/x_{2} ∈ K with x_{1} ∈ O_{K}, x_{2} ∈ O_{K}^{∗} and gcd(x_{1}, x_{2}) = 1, we denote by h(x) the (naive) height of x, namely h(x) := log(max{|x_{1}|, |x_{2}|}) if K = ℚ and h(x) := max{deg(x_{1}), deg(x_{2})} if K = ℚ(T).

For a given algebraic curve C defined over K we denote by C(K) the set of points of the curve C whose coordinates lie in K.

Let C be the K–definable affine (hyperelliptic) curve of 𝔸^{2}(\overline{K}) of equation y^{2} = f(x), where f ∈ K[x] is a square–free polynomial of degree at least 3. For any point P = (x(P), y(P)) ∈ C(K) we define the (naive) height h(P) of P as h(P) := h(x(P)). Further, if P ∈ ℙ^{2}(\overline{K}) is the point of C at infinity we define h(P) := 0.

Let E be an elliptic curve defined over K and let [n] denote the morphism of multiplication by n in E for any n ∈ ℤ \ {0}. For any point P ∈ E(K) we denote by ĥ(P) the canonical height of P, namely ĥ(P) := lim_{n→∞} 4^{−n}h([2^{n}]P).

For P, Q ∈ E(\overline{K}) let ⟨P, Q⟩ denote the Néron–Tate pairing, namely ⟨P, Q⟩ := \frac{1}{2}(ĥ(P + Q) − ĥ(P) − ĥ(Q)). Let us recall that ⟨ , ⟩ induces a positive–definite bilinear form on E(K)/E(K)_{tors}, where E(K)_{tors} denote the set of K–rational points of torsion of E.

It is well–known (see e.g. [Sil86, 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 c_{E} > 0, depending only on the elliptic curve E, such that the estimate

|ĥ(P) − h(P)| < c_{E}  (1)

holds for any P ∈ E(K). The following result will allow us to make the constant c_{E} explicit (see e.g. [Kna92]):

Lemma 1 Let E be an elliptic curve defined over K and let c_{E} > 0 be a constant satisfying the inequality |h([2]P) − 4h(P)| ≤ c_{E} for any point P ∈ E(K). Then the inequality |ĥ(P) − h(P)| ≤ c_{E}/3 holds for any point P ∈ E(K).

3 Points over ℚ(T)

This section is devoted to the proof of Theorem 2, which determines the set of ℚ(T)–rational points of the genus–2 curve C of equation y^{2} = x^{6}+T x^{4}+T x^{2}+1.

As expressed in the introduction, there are two ℚ(T)–definable morphisms ϕ_{1}, ϕ_{2} : C → E mapping this curve to the elliptic curve E defined over ℚ(T) of equation y^{2} = x^{3} + T x^{2} + T x + 1. In order to determine the set C(ℚ(T)) we first determine the structure of the group E(ℚ(T)).

3.1 The structure of E over ℚ(T)

In order to analyze the group E(ℚ(T)) we need an explicit upper bound of 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 x′ = x + T/3 we transform the elliptic curve E into the elliptic curve E′ defined over ℚ(T) of equation y^{2} = x′^{3} + a′x′ + b′, where a′ := −1/3T (T − 3) and b′ := 1/27(2T + 3)(T − 3)^{2}. Then we have the following result:

Lemma 2 Let notations and assumptions be as above. Then for any rational point P ∈ E′(ℚ(T)) the inequality |ĥ(P) − h(P)| ≤ 3/4 holds.

Proof.– Following [ZS01], let M_{ℚ(T)} denote the usual set of all non–equivalent absolute values over ℚ(T), namely the set of all the absolute values v_{𝔭} := − log | |_{𝔭}, where either 𝔭 = ∞ and |F |_{𝔭} := e^{deg(F)}, or 𝔭 runs over the set of all monic prime elements of ℚ[T], and |F |_{𝔭} := e^{−ord_{𝔭}(F)} denotes the standard 𝔭–adic valuation. For any such absolute value v, let

μ_{v}:= min\{\frac{1}{2}v(a′), \frac{1}{3}v(b′)\},  μ:= − ∑_{v∈M_{ℚ(T)}}μ_{v},
μ_{l}:= \frac{1}{2} ∑_{v∈M_{ℚ(T)}}min\{0, μ_{v}\},  μ_{u}:= \frac{1}{2} ∑_{v∈M_{ℚ(T)}}max\{0, μ_{v}\}.

Then [ZS01, Theorem and Proposition 4] shows that −μ − μ_{u} ≤ ĥ(P) − h(P) ≤ −μ_{l} holds for any P ∈ E′(ℚ(T)).

In our case, the only nonzero values of μ_{v} are obtained at 𝔭 = ∞ and 𝔭 = T − 3, namely μ_{∞} = −1 and μ_{T −3} = 1/2. This shows that μ = 1/2, μ_{l} = −1/2 and μ_{u} = 1/4 hold, and then −3/4 ≤ ĥ(P) − h(P) ≤ 1/2. This proves the lemma.

Now we determine the structure of the group of ℚ(T)–rational points of the elliptic curve E. For this purpose, we are going to apply Shioda’s theory of Mordell–Weil lattices of elliptic surfaces (cf. [Shi90, OS91, Shi91]), which actually allows us to describe the larger group E(\overline{ℚ}(T)).

Following [Shi90], associated to the elliptic curve E we have an elliptic surface f : S → ℙ^{1}(\overline{ℚ}) (the Kodaira–Néron model of E/\overline{ℚ}(T)) whose generic fiber is E. For a given v ∈ ℙ^{1}(\overline{ℚ}) let F_{v} := f^{−1}(v) denote the fiber over v, and let R denote the set of reducible fibers F_{v}. For any v ∈ R, let

F_{v} = Θ_{v,0} + ∑_{i=1}^{m_{v}−1} μ_{v,i}Θ_{v,i},

where Θ_{v,i} (0 ≤ i ≤ m_{v} −1) are the irreducible components of F_{v} occurring with multiplicity μ_{v,i} and Θ_{v,0} is the unique component meeting the zero section.

The global sections of S can be naturally identified with the points of E(\overline{ℚ}(T)), namely a given section s : ℙ^{1}(\overline{ℚ}) → S is identified with its restriction to the generic fiber E, which is a \overline{ℚ}(T)–rational point of E. For a given point P ∈ E(\overline{ℚ}(T)) let (P) denote the prime divisor which is the image of the section P : ℙ^{1}(\overline{ℚ}) → S. With this identification Shioda shows that E(\overline{ℚ}(T)) is isomorphic to NS(S)/T, where NS(S) denotes the Néron–Severi group of S (the group of divisors of S modulo algebraic equivalence) and T denotes the subgroup of NS(S) generated by the zero section (O) and all the irreducible components of fibers. In [OS91] there is a complete classification of the possible structures of the group E(\overline{ℚ}(T)) in terms of the root lattices associated with the reducible fibers F_{v}.

There exists a height pairing ⟨ , ⟩ : E(\overline{ℚ}(T)) × E(\overline{ℚ}(T)) → ℚ, which is obtained by embedding E(\overline{ℚ}(T)) into NS(S) ⊗ ℚ. Let us denote by ϕ this embedding. Then we have ker ϕ = E(\overline{ℚ}(T))_{tors}, and using the intersection number as a pairing in NS(S) the height pairing is defined by ⟨P, Q⟩ := −(ϕ(P), ϕ(Q)). In case that the elliptic surface is rational we have

⟨P, P ⟩ = 2 + ((P), O) − ∑_{v∈R} contr_{v}(P),  (2)

where the possible terms contr_{v}(P) are described in [Shi90] in terms of the root lattice associated to the fiber F_{v}.

Proposition 1 The rank of the abelian group E(\overline{ℚ}(T)) is one and its free part is generated by the point G := (0, 1).

Proof.– Let us observe that the singular fibers of S are given at v = −1, 3, ∞. By applying Tate’s algorithm for the determination of the reduction types of the fiber F_{v} (see [Tat75, Sil94]) we see that the special fibers at v = −1, 3, ∞ are of type I_{1}, III, I_{2}^{∗} respectively. This implies m_{−1} = 1, m_{3} = 2 and m_{∞} = 7 respectively. Therefore, only v = 3, ∞ correspond to reducible fibers. Applying the classification of [OS91] we conclude that E(\overline{ℚ}(T)) ≅ A_{1}^{∗} ⊕ ℤ/2ℤ holds, i.e. E(\overline{ℚ}(T)) has rank 1 and E(\overline{ℚ}(T))_{tors} = ℤ/2ℤ.

Since (−1, 0) is a nontrivial torsion point of E(\overline{ℚ}(T)) we conclude that E(\overline{ℚ}(T))_{tors} = ⟨(−1, 0)⟩ holds.

Let us observe that the elliptic surface associated to the elliptic curve E is rational. Therefore, [Shi90, Theorems 10.8 and 10.10] shows that the group E(\overline{ℚ}(T)) is generated by the points P = (x(P), y(P)) satisfying ((P), O) = 0, and hence of the form x(P) = gT^{2} + aT + b, y(P) = hT^{3} + cT^{2} + dT + e.

From [Shi90, Lemma 5.1] we see that A_{1}^{∗} has a basis consisting of a vector P of (minimal) norm ⟨P, P ⟩ = 1/2. Taking into account that contr_{∞}(P) ∈ {0, 1, 3/2} and contr_{3}(P) ∈ {0, 1/2} holds (see [Shi90]), from formula (2) we conclude that contr_{∞}(P) ≠ 0 holds. Arguing as in [Shi91a] we see that this implies that P must intersect the singular fiber F_{∞} (which is a cusp) at the singular point, namely at (0, 0). We conclude that g = h = 0 holds.

Replacing x(P) = aT +b in the right–hand term of the equation defining the elliptic curve E we see that the term p_{a,b}(T) := (aT +b)^{3} +T (aT +b)^{2} +T (aT + b)+1 is not a square in \overline{ℚ}[T] for a ≠ 0 because it has odd degree. Hence we have a = 0. Furthermore, for b ≠ 0, −1 the polynomial p_{0,b}(T) = T (b^{2} + b) + b^{3} + 1 is not a square. Since b = −1 yields a torsion point we conclude that a = b = 0 is the only possible choice for x(P). This shows that G = (0, ±1) is a generator of the free part of E(\overline{ℚ}(T)).

3.2 The structure of C over ℚ(T): 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 ℚ(T) of equation y^{2} = x^{6} + T x^{4} + T x^{2} + 1. Then we have C(ℚ(T)) = {(0, 1), (0, −1)}.

For this purpose we are going to use a simplified version [Kul99] of the Dem’janenko–Manin’s method [Dem68, Man69] for computing the set of rational points of a given genus–2 curve. Proof.– Let us recall that we have two morphisms ϕ_{1}, ϕ_{2} : C → E mapping the curve C into the elliptic curve E, namely ϕ_{1}(x, y) := (x^{2}, y) and ϕ_{2}(x, y) := (1/x^{2}, y/x^{3}).

As in the proof of Lemma 2 we make the change of variable x′ = x + T/3, which transforms the elliptic curve E into the elliptic curve E′ of equation y^{2} = x′^{3} + a′x′ + b′, where a′ := −1/3T (T − 3) and b′ := 1/27(2T + 3)(T − 3)^{2}. We denote by C′ the genus–2 curve defined over ℚ(T) obtained from C under this change of variables and denote by ϕ′_{1}, ϕ′_{2} : C′ → E′ the corresponding morphisms, namely

\begin{aligned}ϕ′_{1}(x′, y) & := & ((x′ − T/3)^{2} + T/3, y), \\ ϕ′_{2}(x′, y) & := & ((x′ − T/3)^{−2} + T/3, y(x′ − T/3)^{−3}) . \end{aligned}

We claim that for any P ∈ C′(ℚ(T)) the following inequality holds:

|h(ϕ′_{1}(P)) − h(ϕ′_{2}(P))| ≤ 1.  (3)

Indeed, let P be an arbitrary element of C′(ℚ(T)) and let x′(P) = N/D be a reduced representation of x′(P). Then the abscissa of ϕ′_{1}(P) is ((3N − DT)^{2} + 3T D^{2})/(9D^{2}). Observe that ((3N −DT)^{2} +3T D^{2})/(9D^{2}) is a reduced fraction and hence h(ϕ′_{1}(P)) = max{deg((3N − DT)^{2} + 3T D^{2}), deg(9D^{2})} holds. Since the leading coefficients of (3N − DT)^{2} and 3T D^{2} are positive rationals we conclude that deg ((3N −DT)^{2}+3T D^{2})) = max{deg ((3N −DT)^{2}), deg(3T D^{2})} > deg(9D^{2}) holds and then h(ϕ′_{1}(P)) = max{deg ((3N − DT)^{2}), deg(3T D^{2})}. Similarly, we see that the abscissa of ϕ′_{2}(P) is (27D^{2} + T (3N − DT)^{2})/(3(3N − DT)^{2}) and h(ϕ′_{2}(P)) = max{deg(27D^{2}), deg (T (3N − DT)^{2})} holds.

Let a := deg(D), b := deg(3N − DT). Then we have h(ϕ′_{1}(P)) = max{2a + 1, 2b} and h(ϕ′_{2}(P)) = max{2a, 2b+1}, which immediately implies estimate (3). This completes the proof of our claim.

Proposition 1 asserts that the abelian group E′(ℚ(T)) has rank 1 and G′ := (T/3, 1) is a generator of its free part. Then for any point P ∈ C′(ℚ(T)) there exist integers n, m and points T_{1}, T_{2} ∈ E′(ℚ(T))_{tors} satisfying the identities ϕ′_{1}(P) = [n]G′ + T_{1} and ϕ′_{2}(P) = [m]G′ + T_{2}. Then we have

ĥ(ϕ′_{1}(P)) = n^{2}ĥ(G′), ĥ(ϕ′_{2}(P)) = m^{2}ĥ(G′).  (4)

Hence, combining identity (3) and Lemma 2 we obtain the following estimate:

\begin{aligned}|ĥ(ϕ′_{1}(P)) − ĥ(ϕ′_{2}(P))| & ≤ & |ĥ(ϕ′_{1}(P)) − h(ϕ′_{1}(P))| + |ĥ(ϕ′_{2}(P)) − h(ϕ′_{2}(P))| \\ & & +|h(ϕ′_{1}(P)) − h(ϕ′_{2}(P))| \\ & ≤ & 2 · 3/4 + 1 = 5/2. \end{aligned}(5)

Let us suppose first that ϕ′_{1}(P) ± ϕ′_{2}(P) ∉ E′(ℚ(T))_{tors} holds. Then m^{2} − n^{2} ≠ 0 and equations (4) and (5) imply ĥ(G′)|m^{2} − n^{2}| < 5/2. Taking into account that h([5]G′) = 15 holds, from Lemma 2 we obtain the estimate ĥ(G′) ≥ 1/2. Therefore, we have min{|n|, |m|} < 5/2 and hence

n, m ∈ {0, ±1, ±2}.  (6)

A direct computation shows that the only ℚ(T)–rational points of C′ satisfying the condition ϕ′_{1}(P) ± ϕ′_{2}(P) ∉ E′(ℚ(T))_{tors} are {(T/3, 1), (T/3, −1)}. We conclude that the only ℚ(T)–rational points of C satisfying the condition ϕ_{1}(P) ± ϕ_{2}(P) ∉ E(ℚ(T))_{tors} are {(0, 1), (0, −1)}.

On the other hand, suppose now that ϕ_{1}(P) ± ϕ_{2}(P) ∈ E(ℚ(T))_{tors} = {O_{E}, (−1, 0)} is satisfied, where O_{E} denotes the zero element of the group E(ℚ(T)). We have that (ϕ_{1} + ϕ_{2})(x, y) = (f_{+}(x), yg_{+}(x)) and (ϕ_{1} − ϕ_{2})(x, y) = (f_{−}(x), yg_{−}(x)), where

f_{+}(x) = \frac{−2x^{3} − 3x^{2} − 2x + T x^{2}}{(x^{4} + 2x^{3} + 2x^{2} + 2x + 1)}, f_{−}(x) = \frac{2x^{3} − 3x^{2} + 2x + T x^{2}}{(x^{4} − 2x^{3} + 2x^{2} − 2x + 1)}.

From the expression of f_{+} and f_{−} we easily conclude that there do not exist points P ∈ C(ℚ(T)) for which ϕ_{1}(P) ± ϕ_{2}(P) ∈ {O_{E}, (−1, 0)} holds. Therefore, the image of the morphisms ϕ_{1}, ϕ_{2} is contained in the set {(0, 1), (0, −1)}. In particular we see that x(P) = 0 holds for any point P ∈ C(ℚ(T)). This shows that C(ℚ(T)) = {(0, 1), (0, −1)} and completes the proof of Theorem 2.

4 Points over ℚ

Let t ∈ ℚ and let C_{t} be the curve of equation y^{2} = x^{6} + tx^{4} + tx^{2} + 1. The purpose of this section is to analyze the arithmetic structure of the curve C_{t}. For this purpose we first determine the arithmetic structure of the elliptic curve E_{t} of equation y^{2} = x^{3} + tx^{2} + tx + 1.

4.1 Explicit bounds

In this section we obtain an explicit upper bound on the height h(P) of any point P ∈ E_{t}(ℚ) 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. [Sil90] and [ZS01]. Nevertheless the following explicit estimate gives better bounds in this case, which allows us to significantly reduce the subsequent computational effort.

Lemma 3 Let t ∈ ℚ. Then for any ℚ–rational point P of the elliptic curve E_{t} the following estimate holds:

|ĥ(P) − h(P)| ≤ \frac{5h(t) + log(1314)}{3}.

Proof.– Let t := b/a and let P be a point of E_{b/a}(ℚ). Let us suppose first that P is not a 2–torsion point. This implies that x(P) does not cancel the 2–division polynomial x^{3} + (b/a)x^{2} + (b/a)x + 1. Then the x–coordinate of the point [2]P is given by the expression

x([2]P) = \frac{a^{2}x(P)^{4} − 2abx(P)^{2} − 8a^{2}x(P) − 4ab + b^{2}}{4a(ax(P)^{3} + bx(P)^{2} + bx(P) + a)} .  (7)

Let us write x(P) := p/q, where p and q are coprime integers. Then we have h(P ) = max{log |p|, log |q|}. Rewriting the identity (7) in terms of p and q we obtain

x([2]P) = \frac{a^{2}p^{4} − 2abp^{2}q^{2} − 8a^{2}pq^{3} + (b^{2} − 4ab)q^{4}}{4qa(ap^{3} + bp^{2}q + bpq^{2} + aq^{3})} .

Let N := a^{2}p^{4} − 2abp^{2}q^{2} − 8a^{2}pq^{3} + (b^{2} − 4ab)q^{4} and D := 4qa(ap^{3} + bp^{2}q + bpq^{2} + aq^{3}) denote the numerator and denominator of the above expression. Then we have the estimates

|N| ≤ (|a|^{2} + 2|ab| + 8|a|^{2} + |b^{2} − 4ab|) max{|p|, |q|}^{4}
≤ 16 max{|a|, |b|}^{2} max{|p|, |q|}^{4},
|D| ≤ 4(|a|^{2} + |ba| + |ba| + |a|^{2}) max{|p|, |q|}^{4}
≤ 16 max{|a|, |b|}^{2} max{|p|, |q|}^{4}.

This yields

h(x([2]P)) ≤ 4h(x(P)) + 2 max{log |a|, log |b|} + log 16.  (8)

Following the proof of [Kna92, Proposition 4.12], let C_{N}, C_{D}, C′_{N}, C′_{D} be integers of minimal height satisfying the Bézout identities

C_{N}N + C_{D}D = Ca^{3}p^{7}, C′_{N}N + C′_{D}D = Cq^{7},  (9)

where C := 108a^{4} − 72a^{2}b^{2} + 32ab^{3} − 4b^{4}. By a direct computation we obtain the following estimates:

|C_{N}| ≤ 664 max{|a|, |b|}^{5} max{|p|, |q|}^{3},
|C_{D}| ≤ 650 max{|a|, |b|}^{5} max{|p|, |q|}^{3},
|C′_{N}| ≤ 40 max{|a|, |b|}^{2} max{|p|, |q|}^{3},
|C′_{D}| ≤ 38 max{|a|, |b|}^{2} max{|p|, |q|}^{3}.

This implies

|p|^{7} ≤ \frac{1314 max\{|a|, |b|\}^{5} max\{|p|, |q|\}^{3} max\{|N|, |D|\}}{|C||a^{3}|},  (10)
|q|^{7} ≤ \frac{78 max\{|a|, |b|\}^{2} max\{|p|, |q|\}^{3} max\{|N|, |D|\}}{|C|}.  (11)

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 Ca^{3}p^{7} and Cq^{7}, i.e. g divides Ca^{3}. Let n := N/g and d := D/g. Then we have

N = ng ≤ nCa^{3}, D = dg ≤ dCa^{3}.

Combining these estimates with inequalities (10) and (11) we obtain

|p|^{7} ≤ 1314 max{|a|, |b|}^{5} max{|p|, |q|}^{3} max{|n|, |d|},
|q|^{7} ≤ 78 max{|a|, |b|}^{5} max{|p|, |q|}^{3} max{|n|, |d|},
max{|p|^{7}, |q|^{7}} ≤ 1314 max{|a|, |b|}^{5} max{|p|, |q|}^{3} max{|n|, |d|}.

(12)

Since n and d are coprime, h(x([2]P)) =h(N/D) =h(n/d) =max{log|n|, log|d|}. Taking logarithms in inequality (12) we obtain

4h(x(P)) ≤ h(x([2]P)) + 5 max{log |a|, log |b|} + log(1314).

Combining this estimate with inequality (8) we deduce the following estimate

|h([2]P) − 4h(P)| ≤ 5 max{log |a|, log |b|} + log(1314).  (13)

Let now P ∈ E(ℚ) be a 2–torsion point. Then x(P) is a root of the polynomial x^{3} + (b/a)x^{2} + (b/a)x + 1. We easily conclude that h(x(P)) ≤ max{log |a|, log |b|} + 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 C_{t} we are going to follow Dem’janenko–Manin’s method [Dem68, Man69, Cas68]. For this purpose we consider the morphisms ϕ_{1}, ϕ_{2} : C_{t} → E_{t} defined by

ϕ_{1}(x, y) := (x^{2}, y), ϕ_{2}(x, y) := (\frac{1}{x^{2}} , \frac{y}{x^{3}} ) .

The application of Dem’janenko –Manin’s method requires an estimate on the difference h(ϕ_{1}(P) + ϕ_{2}(P)) − 4h(P) for any P ∈ C_{t}(ℚ), which is the content of our next result.

Lemma 4 With notations and assumptions as above, for any point P ∈ C_{t}(ℚ) the following inequality holds:

|h(ϕ_{1}(P) + ϕ_{2}(P)) − 4h(P)| ≤ 2h(t) + log(62).

Proof.– Let t := b/a and let P := (x(P), y(P)) be a ℚ–rational point of the curve C_{t}. Suppose first that x(P) = −1. Then ϕ_{1}(P) = −ϕ_{2}(P) and h(P) = 0. We conclude that the statement of Lemma 4 holds in this case.

Suppose now that x(P) ≠ −1 holds. Then we have

x(ϕ_{1}(P) + ϕ_{2}(P)) = \frac{−2ax(P)^{3} + (b − 3a)x(P)^{2} − 2ax(P)}{ax(P)^{4} + 2ax(P)^{3} + 2ax(P)^{2} + 2ax(P) + a}.  (14)

Let us write x(P) = p/q, where p and q are coprime integers. Rewriting identity (14) in terms of p and q we obtain

x(ϕ_{1}(P) + ϕ_{2}(P)) = \frac{−2ap^{3}q + (b − 3a)p^{2}q^{2} − 2apq^{3}}{ap^{4} + 2ap^{3}q + 2ap^{2}q^{2} + 2apq^{3} + aq^{4}} .

Let N := −2ap^{3}q + (b − 3a)p^{2}q^{2} − 2apq^{3} and D := ap^{4} + 2ap^{3}q + 2ap^{2}q^{2} + 2apq^{3} + aq^{4}. Then x(ϕ_{1}(P) + ϕ_{2}(P)) = N/D and we have the estimates

|N| ≤ (2|a| + |b − 3a| + 2|a|) max{|p|, |q|}^{4}
≤ 8 max{|a|, |b|} max{|p|, |q|}^{4},
|D| ≤ (|a| + 2|a| + 2|a| + 2|a| + |a|) max{|p|, |q|}^{4}
≤ 8 max{|a|, |b|} max{|p|, |q|}^{4}.

This implies

h(ϕ_{1}(P) + ϕ_{2}(P)) ≤ 4h(P) + max{log |a|, log |b|} + log 8.  (15)

In order to prove the converse inequality, let C_{N}, C_{D}, C′_{N}, C′_{D} be integers of minimal height satisfying the Bézout identities:

C_{N}N + C_{D}D = Cp^{7}, C′_{N}N + C′_{D}D = Cq^{7},

where C := 3a^{3} + 2a^{2}b − ab^{2}. By a direct computation we obtain the estimates

|C_{N}| ≤ 28 max{|a|, |b|}^{2} max{|p|, |q|}^{3},
|C_{D}| ≤ 34 max{|a|, |b|}^{2} max{|p|, |q|}^{3},
|C′_{N}| ≤ 28 max{|a|, |b|}^{2} max{|p|, |q|}^{3},
|C′_{D}| ≤ 34 max{|a|, |b|}^{2} max{|p|, |q|}^{3}.

Therefore we have

max\{|p|^{7}, |q|^{7}\} ≤ \frac{62 max\{|a|, |b|\}^{2} max\{|p|, |q|\}^{3} max\{|N|, |D|\}}{C}.

Let g be the gcd of N and D. Then g divides Cp^{7} and Cq^{7}. Since p and q are coprime, we conclude that g divides C. Let n, d be the integers such that N = ng and D = dg. Then we have

max{|p|^{7}, |q|^{7}} ≤ 62 max{|a|, |b|}^{2} max{|p|, |q|}^{3} max{|n|, |d|}.

Since n and d are coprime we see that h(x(ϕ_{1}(P) + ϕ_{2}(P))) = h(N/D) = max{|n|, |d|} holds. Therefore, taking logarithms in the previous inequality we deduce the following estimate:

4h(P) ≤ h(ϕ_{1}(P) + ϕ_{2}(P)) + 2 max log{|a|, |b|} + log(62).

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 C_{t}(ℚ).

Theorem 5 Let t be a rational number such that the elliptic curve E_{t} has rank 1 over ℚ. Then for any point P ∈ C_{t}(ℚ) the following estimate holds:

h(P) ≤ \frac{7h(t) + log(81468)}{2}.

Proof.– Let ϕ_{1}, ϕ_{2} : C_{t} → E_{t} be the morphisms ϕ_{1}(x, y) := (x^{2}, y) and ϕ_{2}(x, y) := (1/x^{2}, y/x^{3}) previously introduced. Let P be a fixed point of C_{t}(ℚ). Following the Dem’janenko–Manin’s method we introduce the matrix Ĥ ∈ ℂ^{2×2} defined in the following way:

Ĥ := ( \begin{matrix}ĥ([2]ϕ_{1}(P)) − 2ĥ(ϕ_{1}(P)) & ĥ(ϕ_{1}(P) + ϕ_{2}(P))− \\ & −ĥ(ϕ_{1}(P)) − ĥ(ϕ_{2}(P)) \\ ĥ(ϕ_{1}(P) + ϕ_{2}(P))− & ĥ([2]ϕ_{2}(P)) − 2ĥ(ϕ_{2}(P)) \\ −ĥ(ϕ_{1}(P)) − ĥ(ϕ_{2}(P)) \end{matrix} ) .

Since the elliptic curve E_{t} has rank 1 we have that the points ϕ_{1}(P), ϕ_{2}(P) ∈ E_{t}(ℚ) are ℤ–linear dependent. Therefore, from the positive–definiteness of the Néron–Tate pairing on E_{t}(ℚ)/E_{t}(ℚ)_{tors} we conclude that the matrix Ĥ is singular. Let us observe that Ĥ can be rewritten as:

Ĥ := ( \begin{matrix}2ĥ(ϕ_{1}(P)) & ĥ(ϕ_{1}(P) + ϕ_{2}(P))− \\ & −ĥ(ϕ_{1}(P)) − ĥ(ϕ_{2}(P)) \\ ĥ(ϕ_{1}(P) + ϕ_{2}(P))− & 2ĥ(ϕ_{2}(P)) \\ −ĥ(ϕ_{1}(P)) − ĥ(ϕ_{2}(P)) \end{matrix} ) .

Let H ∈ ℂ^{2×2} be the following matrix:

H := ( \begin{matrix}2h(ϕ_{1}(P)) & h(ϕ_{1}(P) + ϕ_{2}(P))− \\ & −h(ϕ_{1}(P)) − h(ϕ_{2}(P)) \\ h(ϕ_{1}(P) + ϕ_{2}(P))− & 2h(ϕ_{2}(P)) \\ −h(ϕ_{1}(P)) − h(ϕ_{2}(P)) \end{matrix} ) .

From Lemma 3 we have the estimates:

\begin{aligned}|h(ϕ_{i}(P)) − ĥ(ϕ_{i}(P))| & < & \frac{5h(t) + log(1314)}{3}, (i = 1, 2) \\ |h(ϕ_{1}(P) + ϕ_{2}(P)) − ĥ(ϕ_{1}(P) + ϕ_{2}(P))| & < & \frac{5h(t) + log(1314)}{3}. \end{aligned}

We conclude that the entries of the matrix H − Ĥ are real numbers of absolute value bounded by 5h(t) + log(1314).

From the definition of ϕ_{1}, ϕ_{2} we see that h(ϕ_{1}(P)) = h(ϕ_{2}(P)) = 2h(P) holds. We deduce that H can be expressed as H = K + 4h(P)I, where K is the antidiagonal matrix whose nonzero entries are h(ϕ_{1}(P) + ϕ_{2}(P)) − 4h(P) and I denotes the (2 × 2)–identity matrix. Applying Lemma 4 we conclude that the entries of the matrix K are real numbers of absolute value bounded by 2h(t) + log(62).

Let L := Ĥ − H + K. Then the entries of L are real numbers of absolute value bounded by 7h(t) + log(81468) and the matrix Ĥ can be written as Ĥ = L + 4h(P)I.

For a given matrix M := (m_{i,j})_{1≤i,j≤2} ∈ ℂ^{2×2}, let us denote by ∥M∥ the standard ∞–matrix norm of M. We have ∥M∥ ≤ 2 max{|m_{i,j}| : 1 ≤ i, j ≤ 2}. Assuming without loss of generality that h(P) ≠ 0, we see that the matrix (4h(P))^{−1}L + I = (4h(P))^{−1}Ĥ is singular. This implies ∥(4h(P))^{−1}L∥ ≥ 1 (see e.g. [HJ85]). Since the entries of the matrix (4h(P))^{−1}L are real numbers of absolute value bounded by (4h(P))^{−1}(7h(t) + log(81468)) we deduce the estimate h(P) ≤ (7h(t) + log(81468))/2.

From Theorem 5 we shall deduce our first uniform upper bound on the number of rational points of the family of curves {C_{t}}_{t∈ℚ}. For this purpose, we need the following technical result:

Lemma 5 Let G := (0, 1) ∈ E_{t}(ℚ). Then the following estimate holds:

|h([2]G) − 2h(t)| ≤ log(36).

Proof.– Let t := b/a, with a, b ∈ ℤ and gcd(a, b) = 1. The x–coordinate of the point [2]G is given by x([2]G) = (−4ab + b^{2})/4a^{2}. Let N := −4ab + b^{2} and D := 4a^{2}. Then we have |N| ≤ 5 max{|a|, |b|}^{2} and |D| ≤ 4 max{|a|, |b|}^{2}, and thus

h([2]P) ≤ 2 max{log |a|, log |b|} + log(5).  (16)

For the converse inequality, let C_{N}, C_{D}, C′_{N}, C′_{D} be integers of minimal height satisfying the Bézout identities

C_{N}N + C_{D}D = 4a^{2}, C′_{N}N + C′_{D}D = b^{3}.

By a direct computation we obtain the estimates

4|a|^{2} ≤ |D|, |b|^{3} ≤ (5 + 4) max{|a|, |b|} max{|N|, |D|}.

This implies that max{|a|, |b|}^{2} ≤ 9 max{|N|, |D|} holds. Therefore, we have

2 max{log |a|, log |b|} ≤ log(9) + max{log |D|, log |N|}.

Let g be the gcd of N and D and let n := N/g, d := D/g. Then g divides 4a^{2} and b^{3}, and hence divides 4. This implies

2 max{log |a|, log |b|} ≤ log(36) + max{log |d|, log |n|}.

Since n and d are coprime, the above inequality may be rewritten as

2 max{log |a|, log |b|} ≤ h([2]P) + log(36).

Combining this estimate with estimate (16) completes the proof of the lemma.

Let P ⊂ ℚ be the set of values t for which the elliptic curve E_{t} has rank 1 over ℚ and G := (0, 1) is a generator of the free part of the group E_{t}(ℚ). In Section 5 we discuss in a statistical sense how many natural numbers belong to the set P. We have the following result concerning the family of curves {C_{t}}_{t∈P}:

Corollary 1 There exists N ∈ ℕ such that for any t ∈ P we have

#C_{t}(ℚ) ≤ N.

Proof.– Let t ∈ P, let G := (0, 1) ∈ E_{t} and let us fix a point P ∈ C_{t}(ℚ). Let ϕ_{1} : C_{t} → E_{t} be the morphism defined by ϕ_{1}(x, y) := (x^{2}, y). Then there exists n ∈ ℕ and T ∈ E_{t}(ℚ)_{tors} such that ϕ_{1}(P) = [n]G + T holds. Then we have ĥ(ϕ_{1}(P)) = n^{2}ĥ(G).

First we obtain a lower bound for the quantity ĥ(G). From Lemma 3 we have the estimate

ĥ([2]G) ≥ h([2]G) − \frac{5}{3}h(t) − \frac{log(1314)}{3}.

Lemma 5 shows that h([2]G) ≥ 2h(t) − log(36) holds. Therefore, taking into account the identity 4ĥ(G) = ĥ([2]G) and the estimate log(61305984) < 17.94 we obtain the lower bound

ĥ(G) ≥ \frac{h(t) − 17.94}{12}.  (17)

We now estimate the quantity ĥ(ϕ_{1}(P)). On one hand, estimate (13) implies ĥ(ϕ_{1}(P)) − h(ϕ_{1}(P)) ≤ 5h(t)/3 + log(1314)/3. On the other hand, Theorem 5 yields the estimate h(ϕ_{1}(P)) = 2h(P) ≤ 7h(t) + log(81468). Putting together these estimates we obtain

ĥ(ϕ_{1}(P)) ≤ \frac{26}{3}h(t) + 13.71.  (18)

Let t ∈ P satisfy the condition h(t) > 18.94. Then estimate (17) implies ĥ(G)^{−1} ≤ 12(h(t) − 17.94)^{−1}, from which we deduce

n^{2} ≤ 104 \frac{h(t) + 1.59}{h(t) − 17.94}.  (19)

Since the right–hand side of the last estimate is a bounded quantity for any t ∈ ℚ with h(t) > 18.94, we conclude that the cardinality of the set C_{t}(ℚ) is uniformly bounded in the set of values t ∈ P with h(t) > 18.94. On the other hand, the set of values t ∈ ℚ such that h(t) ≤ 18.94 holds is finite. Hence the cardinality of the set C_{t}(ℚ) is uniformly bounded in the set of values t ∈ ℚ with h(t) ≤ 18.94. This concludes the proof of the corollary.

Remark 1 From (19) we easily conclude that for all but finitely many t ∈ P the estimate n ≤ 10 holds.

4.2 The structure of C_{t}(ℚ)

In this section we prove Theorem 3, which determines the arithmetic structure of the curve C_{t} for all but finitely many values t ∈ P, 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}(ℚ).

4.2.1 The torsion subgroup of E_{t}(ℚ)

In order to determine the group C_{t}(ℚ) we first describe the torsion group E_{t}(ℚ)_{tors}. This is the subject of the following proposition.

Proposition 2 For all but finitely many t ∈ ℚ the following assertions hold:

(i) if there exists u ∈ ℚ \ {0, 1, −1} such that t = −(u^{2} − u + 1)/u holds, then

E_{t}(ℚ)_{tors} = \{O_{E_{t}}, (−1, 0), (u, 0), (\frac{1}{u}, 0)\} ,

all points having order 2.

(ii) Otherwise, we have

E_{t}(ℚ)_{tors} := {O_{E_{t}}, (−1, 0)}.

Proof.– Mazur’s Theorem [Maz78] asserts that the torsion subgroup of E_{t}(ℚ) is isomorphic to one of following groups:

• ℤ/mℤ, with 1 ≤ m ≤ 10 or m = 12;

• ℤ/2ℤ × ℤ/2mℤ, with 1 ≤ m ≤ 4.

The point P_{0} := (−1, 0) ∈ E_{t}(ℚ) is a torsion point of order 2. This restricts the choices for the torsion subgroup of E_{t}(ℚ) to ℤ/mℤ with m ∈ {2, 4, 6, 8, 10, 12} and ℤ/2ℤ × ℤ/mℤ with m ∈ {1, 2, 3, 4}. The following lemma restricts further the possible torsion subgroups.

Lemma 6 For all but finitely many t ∈ ℚ the torsion subgroup E_{t}(ℚ)_{tors} of the group E_{t}(ℚ) is isomorphic to ℤ/2ℤ or ℤ/2ℤ × ℤ/2ℤ.

Proof.– Suppose that the torsion group E_{t}(ℚ)_{tors} is not isomorphic to one of the groups ℤ/2ℤ or ℤ/2ℤ×ℤ/2ℤ. Then, the above remarks show that E_{t}(ℚ)_{tors} 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 t ∈ ℚ such that there exists a torsion point of E_{t}(ℚ) of order i is finite.

We sketch the strategy of the proof of the general case and detail the computations in the case i = 3.

Let P := (x(P), y(P)) be a point in E_{t}(ℚ)_{tors}. Then P is an i–torsion point if and only if the i–torsion polynomial p_{i}(t, x) of the elliptic curve E_{t}(ℚ) vanishes in x(P). A direct computation shows that for any i ∈ {3, 4, 5} the equation p_{i}(t, x) = 0 defines genus–0 curve C^{(i)}. Let x = v_{1}^{(i)}(u), t := v_{2}^{(i)}(u) be a parametrization of the curve C^{(i)}, where v_{1}^{(i)}, v_{2}^{(i)} are suitable rational functions of ℚ(u). Replacing this parametrization in the equation y^{2} = x^{3} + tx^{2} + tx + 1 of the elliptic curve E_{t} we obtain a plane curve y^{2} = v^{(i)}(u) which is an elliptic curve of rank 0. This implies that there exists a finite set of ℚ–rational points (u, y) satisfying the equation y^{2} = v^{(i)}(u) and thus a finite set of ℚ–rational points (t, x) satisfying the equation p_{i}(t, x) = 0. Therefore the set of points (x(P), y(P), t) ∈ ℚ^{3} such that P := (x(P), y(P)) is a torsion point of order i of the curve E_{t} is finite. We conclude that set of values t ∈ ℚ for which the curve E_{t} has torsion points of order i is finite.

Now we detail the computations for the case i := 3. In this case the 3– division polynomial is p_{3}(t, x) := 3x^{4} +4tx^{3} +6tx^{2} +12x−t^{2} +4t. The equation p_{3}(x, t) = 0 defines a plane curve of genus 0 which can be parametrized as follows:

x = \frac{(−4 + 3u)(u + 4)}{16u}, t = −\frac{(−4 + 3u)(3u^{3} − 12u^{2} + 144u − 64)}{64u^{3}}.

Replacing this parametrization in the equation y^{2} = x^{3} + tx^{2} + tx + 1 defining the elliptic curve E_{t} we obtain the plane curve

y^{2} = \frac{(u − 4)^{2}(3u^{2} + 24u − 16)^{3}}{16384u^{5}} .  (20)

Making the change of variables y = (u − 4)(3u^{2} + 24u − 16)Y/128u^{3} we see that the non-zero rational solutions of (20) are in bijection with the rational solutions of the curve Y^{2} = 3u^{3} + 24u^{2} − 16u. Taking into account that this is an elliptic of rank 0 over ℚ finishes the proof of our assertion in the case i = 3.

Now we can complete the proof of Proposition 2. By Lemma 6 for all but a finite set of values t ∈ ℚ the torsion group E_{t}(ℚ)_{tors} is isomorphic to one of the groups ℤ/2ℤ and ℤ/2ℤ × ℤ/2ℤ. Let us fix a value t ∈ ℚ such that the group E_{t}(ℚ)_{tors} is isomorphic to the group ℤ/2ℤ × ℤ/2ℤ. Then E_{t}(ℚ)_{tors} has three distinct elements of order 2, whose x–coordinates are three distinct rational roots of the polynomial

p_{2,t}(x) := x^{3} + tx^{2} + tx + 1 = (x + 1)(x^{2} + tx − x + 1).

In such a case, there exists a root u ∈ ℚ \ {0, −1, 1} of the polynomial p_{2,t} and hence t = −(u^{2} − u + 1)/u holds (observe that the values u = ±1 make the curve E_{t} singular). We easily conclude that the torsion subgroup of E_{t}(ℚ) is

E_{t}(ℚ)_{tors} = \{O_{E_{t}}, (−1, 0), (u, 0), (\frac{1}{u}, 0)\} .

On the other hand, if the group E_{t}(ℚ)_{tors} is isomorphic to ℤ/2ℤ, taking into account that (−1, 0) is a nontrivial torsion point of E_{t}(ℚ) we conclude that E_{t}(ℚ)_{tors} = {O_{E_{t}}, (−1, 0)} holds. This completes the proof of Proposition 2.

4.2.2 The set C_{t}(ℚ)

Now we are able to prove Theorem 3, which determines the set of ℚ–rational points of the curve C_{t} for all but finitely many values t ∈ P. Theorem 3 For all but finitely many values t ∈ P the following assertions hold:

(i) if there exists v ∈ ℚ such that t = −(v^{4} − v^{2} + 1)/v^{2} holds, then

C_{t}(ℚ) = \{(0, 1), (0, −1), (v, 0), (−v, 0), (\frac{1}{v} , 0), ( − \frac{1}{v} , 0)\} .

(ii) Otherwise, we have

C_{t}(ℚ) = {(0, 1), (0, −1)}.

Proof.– Let t ∈ ℚ and let as before ϕ_{1}, ϕ_{2} : C_{t} → E_{t} denote the morphisms defined by ϕ_{1}(x, y) := (x^{2}, y) and ϕ_{2}(x, y) := (1/x^{2}, y/x^{3}). Observe that for any point P = (x(P), y(P)) of C_{t}(ℚ) we have ϕ_{1}(P) ∈ E_{t}(ℚ) and ϕ_{2}(P) ∈ E_{t}(ℚ). Corollary 1 and Remark 1 show that for all but a finite set of values t ∈ P the points ϕ_{1}(P) and ϕ_{2}(P) can be expressed as ϕ_{1}(P) = [n_{1}](0, 1) + T_{1} and ϕ_{2}(P) = [n_{2}](0, 1) + T_{2}, with |n_{1}|, |n_{2}| ≤ 10 and T_{1}, T_{2} ∈ E_{t}(ℚ)_{tors}.

Let us fix for the moment an integer n and a torsion point T := (t_{1}, t_{2}) of E_{t}. Then the x–coordinate of the point [n](0, 1) + T ∈ E_{t}(ℚ) can be expressed as a rational function in the value t, which we denote by F_{n,T} (t). We shall see that for any point P ∈ C_{t}(ℚ) the definition of the morphisms ϕ_{1}, ϕ_{2} imply that there exist T_{1}, T_{2} ∈ E_{t}(ℚ)_{tors} such that the condition F_{n_{1},T_{1}}(t)F_{n_{2},T_{2}}(t) = 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 t ∈ P and let us suppose that there exists v ∈ ℚ such that t = −(v^{4} − v^{2} + 1)/v^{2}. Letting u := v^{2} we see that there exists u ∈ ℚ \ {0, 1, −1} for which t = −(u^{2} − u + 1)/u holds. Then Proposition 2(i) shows that the torsion subgroup of E_{t}(ℚ) is given by E_{t}(ℚ)_{tors} = \{O_{E_{t}}, (−1, 0), (u, 0), (\frac{1}{u}, 0)\} =: \{T_{1}, T_{2}, T_{3}, T_{4}\}, all points having order 2. Then any point T ∈ E_{t}(ℚ)_{tors} has order at most 2 and we have that for any n ∈ ℤ the x–coordinates of the points [n](0, 1) + T and [−n](0, 1) + T agree. Therefore, in order to determine which are the possible x–coordinates of the image of a point P ∈ C_{t}(ℚ) we may assume without loss of generality that n ≥ 0 holds.

For 1 ≤ i ≤ 4 and 0 ≤ n ≤ 10, let F_{n,i}(u) denote the rational function which represents the x–coordinate of the point [n](0, 1) + T_{i}. Let P := (x(P), y(P)) be a point of C_{t}(ℚ). Then Proposition 2(i) and Remark 1 show that for all but finitely many values t ∈ P we have that x(P) and u satisfy the condition:

x(P)^{2} = F_{n_{1},j_{1}}(u), \frac{1}{x(P)^{2}} = F_{n_{2},j_{2}}(u),  (21)

with 0 ≤ n_{1}, n_{2} ≤ 10 and j_{1}, j_{2} ∈ {1, 2, 3, 4}. Let us observe that the cases n_{1} = 0, j_{1} = 1 and n_{2} = 0, j_{2} = 1 cannot arise because the point O_{E_{t}} = [0](0, 1) does not belong to the affine part of the curve E_{t}. On the other hand, the cases n_{1} = j_{1} = 1 and n_{2} = j_{2} = 1 yield the point (0, 1) = [1](0, 1), which is the image of the points (0, ±1) ∈ C_{t}(ℚ). Finally, the cases n_{1} = 0, j_{1} = 2 and n_{2} = 0, j_{2} = 2 cannot arise because the x–coordinate of the point [0](0, 1) + (−1, 0) = (−1, 0) is not a square in ℚ. In all the remaining cases (21) shows that the equation

F_{n_{1},j_{1}}(u)F_{n_{2},j_{2}}(u) = 1  (22)

holds. A direct computation shows that this identity is satisfied for all the values u ∈ ℚ if and only if n_{1} = n_{2} = 0 and j_{1} = 3, j_{2} = 4 or j_{1} = 4, j_{2} = 3 hold.

In all the other cases F_{n_{1},j_{1}}(u)F_{n_{2},j_{2}}(u) − 1 is a nonzero rational function which vanishes in a finite set values u ∈ ℚ. Since there are only a finite set of possible choices for the integers n_{1}, n_{2}, j_{1}, j_{2}, we conclude that for all but finite many values u ∈ ℚ the identity (22) will not be satisfied unless n_{1} = n_{2} = 0 and j_{1} = 3, j_{2} = 4 or j_{1} = 4, j_{2} = 3 hold. In this latter case the conditions x^{2} = F_{0,3}(u) = u or x^{2} = F_{0,4}(u) = u are satisfied if and only if u is a square in ℚ, which holds true since by assumption u = v^{2}. Taking into account that that the fiber of the set {(u, 0), (1/u, 0)} under the morphisms ϕ_{1}, ϕ_{2} is the set {(±v, 0), (±1/v, 0)} we easily conclude the statement of Theorem 3(i).

Proof of Theorem 3(ii). Now we have that there does not exist v ∈ ℚ such that t = −(v^{4} −v^{2} +1)/v^{2}. If there exists u ∈ ℚ for which t = −(u^{2} −u+1)/u holds, the arguments of the proof of Theorem 3(i) show that C_{t}(ℚ) = {(0, 1), (0, −1)} holds. Therefore, we may assume without loss of generality that that there does not exist u ∈ ℚ such that t = −(u^{2} − u + 1)/u holds. Then Proposition 2(ii) shows that E_{t}(ℚ)_{tors} = {O_{E_{t}}, (−1, 0)} holds. Let us fix n ∈ ℤ. Then there exist rational functions F_{n,1}, F_{n,2} ∈ ℚ(t) which represent the x–coordinate of the points [n](0, 1) and [n](0, 1) + (−1, 0) respectively. Arguing as before we conclude that without loss of generality we may assume that n ≥ 0 holds.

Let P := (x(P), y(P)) be a point in C_{t}(ℚ). From Remark 1 we deduce that x(P) and t satisfy the relation:

x^{2}(P) = F_{n_{1},j_{1}}(t), \frac{1}{x^{2}(P)} = F_{n_{2},j_{2}}(t)  (23)

with 0 ≤ n_{1}, n_{2} ≤ 10 and j_{1}, j_{2} ∈ {1, 2}. We observe that the cases n_{1} = 0, j_{1} = 1 and n_{2} = 0, j_{2} = 1 do not yield points of C_{t}(ℚ), because the point [0](0, 1) does not belong to the the affine part of the elliptic curve E_{t}. On the other hand, the cases n_{1} = 0, j_{1} = 2 and n_{2} = 0, j_{2} = 2 do not yield points of C_{t}(ℚ), because the x–coordinate of the point [0](0, 1) + (−1, 0) = (−1, 0) is not a square in ℚ. Finally, in the case n_{1} = j_{1} = 1 we have the point (0, 1) ∈ E_{t}(ℚ), whose ϕ_{1}–fiber is the set {(0, 1), (0, −1)} for any t ∈ ℚ.

In all the remaining cases (23) implies F_{n_{1},j_{1}}(t)F_{n_{2},j_{2}}(t) = 1. Furthermore, in all these cases F_{n_{1},j_{1}}(t)F_{n_{2},j_{2}}(t) − 1 is a nonzero element of ℚ(t), thus vanishing in a finite set of values t ∈ ℚ. Since there are only a finite set of admissible choices for the integers n_{1}, n_{2}, j_{1}, j_{2} we conclude that for all but a finite set of values t ∈ ℚ the identity C_{t}(ℚ) = {(0, 1), (0, −1)} holds. This concludes the proof of Theorem 3(ii).

5 Experimental and conjectural results

Theorem 3 asserts that the cardinality of the set C_{t}(ℚ) is uniformly bounded in the set of values t ∈ ℚ satisfying the following conditions:

1. The rank of the abelian group E_{t}(ℚ) is 1.

2. (0,1) is a generator of the free part E_{t}(ℚ).

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.

5.1 Rank considerations

Since Theorem 2 shows that conditions 1 and 2 are satisfied by the elliptic curve E defined over ℚ(T), 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. Indeed, J. Cassels and A. Schinzel [CS82] exhibit a rank–0 elliptic curve Ẽ defined over ℚ(T) with the following property: assuming Selmer’s conjecture [Sel54], for any t ∈ ℚ the specialized curve Ẽ_{t} has rank at least 1.

The general question of characterizing the behaviour of the rank of an elliptic curve defined over ℚ(T) under specializations is a difficult problem (see e.g. [Sil85]). Nevertheless there is some numerical experience, as that of S. Fermigier [Fer96] who studies 66918 elliptic curves Ẽ_{t} with t ∈ ℤ, coming from 93 ℚ(T)- definable elliptic curves Ẽ having ranks between 0 and 4 over ℚ(T). S. Fermigier shows that, with a surprising amount of uniformity, the following identity holds:

rank Ẽ_{t}(ℚ) = rank Ẽ(ℚ(T)) + N,

where

N = 0 with probability 32%,
N = 1 with probability 48%,
N = 2 with probability 18%,
N = 3 with probability 2%.

We computed the rank of 284051 elliptic curves E_{t} with h(t) ≤ log(530). We obtain the following results:

rank E_{t}(ℚ) = rank E(ℚ(T)) + N,

where

N = 0 with probability 32.7%,
N = 1 with probability 49.9%,
N = 2 with probability 15.9%,
N = 3 with probability 1.5%.

These figures suggest that condition 1 might hold with a probability of success of approximately 1/3. We refer to [Sil98] 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 E(ℚ(T)), 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}(ℚ).

This problem can be put into a general setting: let Ẽ be a elliptic curve defined over ℚ(T); then for all but finitely many t ∈ ℙ^{1}(ℚ) the specialized curve Ẽ_{t} is an elliptic curve defined over ℚ(T) and we may consider the specialization homomorphism σ_{t} : Ẽ(ℚ(T)) ↦ Ẽ_{t}(ℚ).

In [Sil85], J. Silverman asks whether the image of σ_{t} is divisible in Ẽ_{t}(ℚ) for values t ∈ ℕ, i.e. whether there are points P ∈ Ẽ_{t}(ℚ) such that [n]P ∈ σ_{t}(Ẽ(ℚ(T))) for some integer n ≥ 2 and P ∉ σ_{t}(Ẽ(ℚ(T))) for t ∈ ℕ. Theorems 2 and 3 of [Sil85] give the following result.

Theorem 6 [Sil85] Let notations and assumptions as above. Suppose further that the elliptic curve Ẽ has nonconstant j–invariant. Then the following assertions hold:

(i) The set of values t ∈ ℕ for which σ_{t}(Ẽ(ℚ(T))) is indivisible in Ẽ_{t}(ℚ) has density 1.

(ii) Assuming that Conjecture B is true, there exists an absolute constant C > 0 with the following property : for any t ∈ ℕ and any P ∈ E_{t}(ℚ) for which P ∈ σ_{t}(Ẽ(ℚ(T))) ⊗ ℚ holds, there exists 0 ≤ n < C such that [n]P ∈ σ_{t}(Ẽ(ℚ(T))) holds.

Applying Theorem 6 to the elliptic curve E of equation y^{2} = x^{3}+T x^{2}+T x+1 we obtain the following result:

Corollary 2 Let Q denote the set of values t ∈ ℚ such that the abelian group E_{t}(ℚ) has rank 1 and let R denote the (density 1) set of values t ∈ ℕ for which σ_{t}(E(ℚ(T))) is indivisible in E_{t}(ℚ).

(i) For any t ∈ R ∩ Q, the point (0, 1) generates the free part of E_{t}(ℚ).

(ii) Assuming that Conjecture B is true, there exists C̃ ∈ ℕ such that the following property holds: for any t ∈ ℕ∩Q, if G_{t} is a generator of the free part of E_{t}(ℚ) then there exists n ≤ C̃ such that (0, 1) − [n]G_{t} ∈ E_{t}(ℚ)_{tors} holds.

Proof.– Let σ_{t} : E(ℚ(T)) → E_{t}(ℚ) be the specialization homomorphism of the elliptic curve E. [Sil83] shows that for all but finitely many values t ∈ ℚ the homomorphism σ_{t} is injective. This implies that for all but finitely many values t ∈ ℚ the subgroup of E_{t}(ℚ) generated by the point (0, 1) is a torsion free subgroup of rank 1.

Let t ∈ R ∩ Q and let G_{t} be a generator of the free part of the group E_{t}(ℚ). Then there exist m ∈ ℤ and T ∈ E_{t}(ℚ)_{tors} such that (0, 1) = [m]G_{t} + T holds. Therefore, multiplying this identity by n := 3 · 5 · 7 · 8 · 11 we conclude that [n](0, 1) = [nm]G_{t} holds. Since [nm]G_{t} = [n](0, 1) ∈ σ_{t}(E(ℚ(T))), by the indivisibility of σ_{t}(E(ℚ(T))) we see that G_{t} ∈ σ_{t}(E(ℚ(T))) holds.

Let G ∈ E(ℚ(T)) be such that σ_{t}(G) = G_{t} holds. By Proposition 1 we have G = [s](0, 1) + [s′](−1, 0) with s ∈ ℤ and s′ ∈ {0, 1}. Then we have G_{t} = [s]σ_{t}(0, 1) + [s′]σ_{t}(−1, 0) = [s](0, 1) + [s′](−1, 0). Multiplying this identity by m we have (0, 1) − T = [m]G_{t} = [ms]σ_{t}(0, 1) + [ms′]σ_{t}(−1, 0). We conclude that the point (1 − ms)(0, 1) is a torsion point of E_{t}(ℚ), which implies ms = 1. From this we easily deduce that the point (0, 1) generates the free part of the group E_{t}(ℚ). This shows assertion (i).

For the second assertion, arguing as above we have that there exists m ∈ ℤ \ {0} and T ∈ E_{t}(ℚ)_{tors} such that [m]G_{t} + T = (0, 1) holds. Then we have [mn]G_{t} ∈ σ_{t}(E(ℚ(T))), where n := 3 · 4 · 5 · 7 · 11. If G_{t} ∈ σ_{t}(E(ℚ(T))) and G ∈ E(ℚ(T)) satisfies σ_{t}(G) = G_{t}, then there exists s, s′ ∈ ℤ such that G_{t} = [s](0, 1) + [s′](−1, 0) holds. Arguing as above we conclude that ms = 1, which implies (0, 1) − [m]G_{t} ∈ E_{t}(ℚ)_{tors} with |m| ≤ 1.

Suppose now that G_{t} ∉ σ_{t}(E(ℚ(T))) holds. Then Theorem 6(ii) shows that mn ≤ C′ holds, where C′ is the constant of the statement of Theorem 6(ii) for the curve E. Thus (0, 1) − [m]G_{t} ∈ E_{t}(ℚ)_{tors} with |m| ≤ C′/n. This concludes the proof of assertion (ii).

We experimentally analyzed the density of the set R ∩ Q of values t ∈ ℚ for which the rank of E_{t}(ℚ) is 1 and the point (0, 1) generates the free part of the group E_{t}(ℚ). For this purpose we tested 28469 elliptic curves E_{t} of rank 1 with h(t) ≤ log(280). We found that the point G := (0, 1) ∈ E_{t}(ℚ) is a generator of the free part of E_{t}(ℚ) in 99.4% of these curves.

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 t ∈ ℕ ∩ Q, even in the case that the point (0, 1) ∈ E_{t}(ℚ) does not generate the free part of the group E_{t}(ℚ): Theorem 4 Assuming that Conjecture B is true, for any t ∈ ℕ ∩ Q the cardinality of the set C_{t}(ℚ) is uniformly bounded.

Proof.– Let G_{t} be a generator of the free part of E_{t}(ℚ). Then Corollary 2(ii) shows that there exists n ≤ C such that (0, 1) − [n]G_{t} ∈ E_{t}(ℚ)_{tors} holds, where C is the constant of Corollary 2(ii) . Then we have ĥ(0, 1) ≤ C^{2}ĥ(G_{t}). Moreover, from the proof of Corollary 1 we see that if h(t) > 18.94 holds then ĥ(0, 1)^{−1} ≤ 12(h(t) − 17.94)^{−1} holds. This implies the estimate

\frac{1}{ĥ(G_{t})} ≤ \frac{12C^{2}}{h(t) − 17.94}.  (24)

Let P be a point of C_{t}(ℚ). Then there exist n ∈ ℕ and T ∈ E_{t}(ℚ)_{tors} such that ϕ_{1}(P) = [n]G_{t} + T holds. Hence we have ĥ(ϕ_{1}(P)) = n^{2}ĥ(G_{t}). On the other hand, from the proof of Corollary 1 we deduce the estimate

ĥ(ϕ_{1}(P)) ≤ \frac{26}{3}h(t) + 13.71.  (25)

Let t ∈ ℕ satisfy the condition t > 18. Then estimates (24) and (25) imply

n^{2} ≤ 104C^{2} \frac{t + 1.59}{t − 17.94}.

Since the right–hand side of the last estimate is a bounded quantity for any t ≥ 19, we conclude that the cardinality of the set C_{t}(ℚ) can be uniformly bounded for any t ≥ 19 such that the rank of the group E_{t}(ℚ) is 1. On the other hand, the set of values {1, …, 18} is finite and hence the cardinality of the set C_{t}(ℚ) can be uniformly bounded for all t ∈ {1, …, 18}. This concludes the proof of the theorem.

Acknowledgments. The authors would like to thank Joseph Silverman and Noam Elkies for their corrections and suggestions that greatly improved the contents and presentation of this paper.

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