On the Equation 𝑦²=𝑥³−𝑝𝑞𝑥

Abstract

We consider certain quartic twists of an elliptic curve. We establish the rank of these curves under the Birch and Swinnerton-Dyer conjecture and obtain bounds on the size of Shafarevich-Tate group of these curves. We also establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists.

1. Introduction

In this paper, we investigate certain quartic twists of the elliptic curve and present some of their interesting properties. Specifically, we consider the family of elliptic curves , where with and distinct prime numbers, . These elliptic curves have complex multiplication by . The -torsion point generates the torsion subgroup of the Mordell-Weil group . Our first result concerns the rank of and an interesting valuational property of points in . More specifically we obtain the following.

Theorem 1. Let , where with and distinct prime numbers and . Then the rank of the Mordell-Weil group is less than or equal to 1. If the rank is one, then for every point of which is not in , the -adic and -adic valuations of must have opposite parity. Moreover, under the Birch and Swinnerton-Dyer conjecture, the rank of is one.

The situation where and do not satisfy the congruence condition in Theorem 1 is less clear. Recently Li and Zeng [1] showed, under the Birch and Swinnerton-Dyer conjecture, that, for where and are distinct odd primes, there exists an elliptic curve , where depends on the classes of and modulo 8, such that the elliptic curve has rank 1 and the valuations at and of -coordinate are not equal for odd , where is a generator of the Mordell-Weil group .

By assuming conjectures, in addition to the Birch and Swinnerton-Dyer conjecture, we also obtain the following.

Theorem 2. Let , where with and distinct prime numbers and . Then the following holds under the Birch and Swinnerton-Dyer conjecture, the elliptic curve analog of the Brauer-Siegel theorem, and the Hardy-Littlewood’s F conjecture. For every there are infinitely many with and and prime with , such that , where is the Shafarevich-Tate group of and is the minimal discriminant of .

Let be an elliptic curve over . The naïve height of the elliptic curve is defined to be where and are the -invariants associated to a minimal model of .

Let vary over a family of number fields of a fixed extension degree over . Let , , and denote, respectively, the discriminant, the class number, and the regulator of . The Brauer-Siegel theorem [2] states that if tends to infinity then . The elliptic curve analog of the Brauer-Siegel Theorem [3] asserts that, for a family of elliptic curves defined over a fixed number field such that the height tends to infinity, .

The Hardy and Littlewood conjecture F [4] concerns polynomials of the form , where are integers and is positive. It asserts that if the greatest common divisor of the coefficients is 1 and is not a square, and either or is odd, then asymptotically, the number of primes less than of the form is given by , where is a constant depending only on .

The curve has as a minimal model, with discriminant and naïve height . The elliptic analog of Brauer-Siegel implies that , and Theorem 2 implies that the bound is essentially tight.

We remark that a result of de Weger [5] demonstrates that for every there exist infinitely many elliptic curves of rank 0 with assuming the BSD conjecture in the rank case and a conjecture of Shintani and Shimura that the Riemann hypothesis holds for the Ranking-Selberg zeta function associated to the weight modular form associated to an elliptic curve by the Shintani-Shimura lift.

The result was improved in [6] where it was shown that, for every , there are infinitely many elliptic curves of rank 0 such that assuming the BSD conjecture in the rank case and the conjecture of Shintani and Shimura.

The proof of Theorem 1 is given in Section 2. The proof of Theorem 2 is given in Section 3. By Theorem 1, any point of the Mordell-Weil group which is not in must behave differently with respect to -adic and -adic valuations. This sets the stage for reductions between the problem of factoring integers of the form and the problem of computing nontorsion rational points on . This is discussed in Section 4. In one direction, it is shown that how given a rational nontorsion point , the factors and of can be found in time polynomial in the height of . We note that the cases where have rational points of small height are those that give rise to large Shafarevich-Tate groups (see Section 3).

2. Proof of Theorem 1

This section is devoted to the proof of Theorem 1. Part of the analysis closely follows Sections and X.4.9 from Silverman’s book [7]. First we recall some facts that follow as the exposition in Example X.4.8 and Proposition X.4.9 applies to our situation. We also adopt the notation there.

Let over be the elliptic curve where (the subscript will be dropped when it is clear from the context). Then is isogenous to the elliptic curve via the isogeny . The kernel of consists of and , the identity of . Let be the dual isogeny of .

Let be the set of primes of and . Let that consists of and all primes dividing 2 or . Let denote the completion of with respect to the absolute value associated to . In particular, denotes and for , denotes the -adic numbers. Let be the subgroup of defined as follows:

Now let , where and are odd and distinct primes, and the group is generated by , , and .

Let denote the Weil-Châtelet group of , the group of equivalence classes of homogeneous spaces for over . For each , the corresponding homogeneous spaces and , also referred to as quartics, are given by the equations

Identifying with , we have , under which the -Selmer group can be viewed as a subset of as follows: The -Selmer group has an analogous isomorphism where is replaced by .

There is the well-known exact sequence: where the map is defined through the connecting homomorphism with . Under the isomorphism we have and if .

Similarly there is an injection sending to and to if .

Thus, the images of , the -torsion point of , and in the Selmer groups are given by respectively.

We now restrict our attention to where and and are distinct primes such that . Also swapping and if necessary we have .

Below is a descent analysis on the -isogenous elliptic curves and . We would like to thank an anonymous referee for valuable suggestions, which we adopt here. Our original analysis can be found in [8, Appendix ].

When or , we have an injection induced from the connecting homomorphism sending to and to if . When , we write for the actual image of . By virtue of the exact sequence it follows that consists of with localizations in for all .

Suppose , being equal to , or a place of good reduction then and we find that .

For or , has bad, type III reduction. The group is generated by and the point [7, Table 15.1]. Since both and (the last isomorphism due to the type of reduction being additive) are divisible by , so is the group . It follows that . We recall that it is assumed that , labeled so that . Since and are squares modulo and modulo , respectively, we have illustrated that and .

For to be an element of , it is necessary and sufficient that when localized it maps to a unit in when is and when is a place of good reduction and to an element of when is and . It follows that the Selmer group is .

Next, we consider the group . If then . This fact coupled with reasoning similar to the preceding paragraphs shows that .

From the exact sequences following an analysis similar to that used in proving Proposition X.6.2(c) [7], we obtain where denotes the rank of and is the dimension as a -vector space. In particular, .

When , then . Since it follows that is trivial, so gives an isomorphism . So the points on map onto . Since the image of is , we find that, for any point in but not in , and have opposite parity.

To finish the proof of Theorem 1, we need to argue, on BSD, that the elliptic curves have Mordell-Weil rank . To this end we investigate the zeros of the -series of at . For the curves of interest the global root number can be computed from the formulae in [9] and it equals −1. Evaluating the functional equation at , we have and hence . This implies that ; in other words, the analytic rank .

If ; that is, , then , by a result of Kolyvagin [10]. Alternatively, since and , it follows from the BSD conjecture that .

This completes the proof of Theorem 1.

3. Proof of Theorem 2

The curve has as a minimal model, with discriminant and height . The -torsion point generates the torsion subgroup of the Mordell-Weil group .

Let denote the isogenous curve of , and let represent the homogeneous spaces of .

One way to compute a rational point of is to search for a rational point on the homogeneous spaces: (assuming ) and this gives us a rational point of via the map . For example if has a rational point , where the numerator and denominator of are polynomially bounded in , then a rational point on can be found of canonical height .

We note that in the special case where is a square the curve does have a small rational point . Thus if there are infinitely many such pairs of primes and , then there are infinitely many with with a rational point of height bounded by . In the simplest case taking , the question boils down to whether there are infinitely many primes of the form . The answer is affirmative under Hardy-Littlewood’s F conjecture [4].

Thus assuming Hardy-Littlewood’s F conjecture and BSD, there is a subfamily of infinitely many such that with and prime and such that, for any , , for sufficiently large . Theorem 2 now follows from the elliptic analog of the Brauer-Siegel theorem and the fact that has minimal discriminant and height .

4. Implications on Computational Complexity

In Theorem 1 we argue that any point of the Mordell-Weil group which is not in must behave differently with respect to -adic and -adic valuations. This sets the stage for reductions between the problem of factoring integers of the form and the problem of computing nontorsion rational points on .

We discuss below that how given a rational nontorsion point , the factors and of can be found in time polynomial in the height of .

Let and be the numerator and denominator of , respectively. Let , the naïve logarithmic height of . And let be the naïve logarithmic height of which is .

Suppose we are given , a nontorsion rational point on . A point can be constructed by “halving” using the duplication formula [7, Algorithm 2.3 (d)]. We note that the canonical height of is not greater than that of . From the reasoning above we have . Below we describe how and can be recovered from .

Clearing the denominators of point , we consider , where . (Taking to be will also suffice for our argument but might lead to trickier analysis). We note that the integer can be viewed as the -coordinate of a point on the projective curve .

Since , it follows that computing gives us either or but neither nor , where such that but does not. Moreover, . Suppose , for ; then . The fact that implies and hence . The integer can be found using the usual doubling trick in bits operations. And the running time of the reduction is where is the bit operations to multiply two -bits numbers. Therefore the overall time complexity of the reduction is softly linear in the height of the point for fixed .

Further, suppose that height of the point is a polynomial in the height of the elliptic curve ; then the time complexity of the reduction is polynomial in .

We note that in the special case where is a square the curve does have a small rational point , and finding such a point is easily reduced to factoring . Therefore in this case finding a rational point on and factoring are polynomial time equivalent.

More generally we remark that one of the procedures to compute a rational point of is to search for a rational point on the homogeneous spaces: , (assuming ) and this gives us a rational point of via the map . The knowledge of the two factors and allows us to write down the equation of the homogeneous spaces. Moreover suppose either one of the two homogeneous spaces has a small rational point, say has a rational point where the numerator and denominator of are polynomially bounded in . Then a rational point can be found by exhaustive search (since is so small), and a rational point of can be obtained from such . Consequently in such cases finding a rational point on and factoring are polynomial time equivalent. Note that these are also cases that give rise to large groups in Theorem 2.

In general it is an interesting open question to determine to what extent finding a rational point on and factoring are polynomial time equivalent.

In light of the question of finding a rational point on , it may be interesting to investigate the efficiency of Heegner point computation when restricted to these elliptic curves . Assuming BSD we observe the dependence of the Heegner index on , which may be big (for instance under the Brauer-Siegel Analogue for elliptic curves). It follows that factoring numbers of the form by computing a point in via the Heegner point method would be computationally expensive. We refer to [8] for a detailed discussion.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees of an earlier version of the paper for valuable suggestions. The authors would also like to thank William Stein for providing access to  Sage software system [11] via   http://modular.math.washington.edu/ (funded by National Science Foundation Grant no. DMS-0821725). The second author was supported in part by NSF Grants CCR-0306393 and CNS-0627458.