ON THE FAMILY OF ELLIPTIC CURVES y 2 = x 3 m 2 x+p 2

Transcription

1 Manuscript ON THE FAMILY OF ELLIPTIC CURVES y = x m x+p ABHISHEK JUYAL, SHIV DATT KUMAR Abstract. In this paper we study the torsion subgroup and rank of elliptic curves for the subfamilies of E m,p : y = x m x +p, where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of E m,p (Q) is trivial for both the cases {m, m 0 (mod )} and {m, m 0 (mod ),with gcd(m,p) = }. We also show that given any odd prime p and for any positive integer m with m 0 (mod ) and m (mod ), the lower bound for the rank of E m,p (Q) is. Finally we find curves of rank in this family.. Introduction Brown and Myers [] constructed an infinite family E m : y = x x+m of elliptic curves over Q with quadratic growth of parameter and the rank of the Mordell-Weil group at least two. Antoniewicz [] produced another family of elliptic curves C m : y = x m x + over Q with the rank of Mordell-Weil group at least three. Ekinberg [], in his Ph.D. thesis, studied the families E m and C m and found some subfamilies of E m and C m of high rank. He also considered the family D m : y = x m x+m, and has shown that the Mordell- Weil rank of D m is over Q(m) with generators (m,m),(0,m). Petra Tadić carried out an interesting study of previously mentioned families E m and C m over function fields. In [], Tadić proved that the torsion subgroup ofc m : y = x m x+ over the function fieldc(m) is trivial, and rank of C m is and over the function fields Q(m) and C(m) respectively. Furthermore, Tadić derived a parametrization of C m of rank at least four over the function field Q(a,i,s,n,k), where s = i a i. Tadić [] found a subfamily of elliptic curves E m having rank over the function field Q(a,i,s,n,k,l), where s = i + a. Additionally, Tadić applied the results of [] to prove the existence of two more families; the first one with ranks and over the field of rational functions in four variables and the second one is a family of rank parametrized by an elliptic curve of positive rank. It is of particular interest to specifically study the family:e m,p : y = x m x+p of elliptic curves parametrized by two rational parameters. Work of Brown-Myers and Antoniewicz [, ] is the source of inspiration for the problem of the presented work and methodology developed in this article. The main object of this paper is to prove the following results. Theorem.. Let E m,p : y = x m x+p, 00 Mathematics Subject Classification. G0. Key words and phrases. Elliptic curve, Rank, Torsion Subgroup.

2 ABHISHEK JUYAL, SHIV DATT KUMAR be a family of elliptic curves, where p is a prime number and m is a positive integer. Then if () m 0 (mod ), () m 0 (mod ) and gcd (m,p) =. Theorem.. Let Tors E m,p (Q) = {O} E m,p : y = x m x+p, be a family of elliptic curves, where m is a positive integer such that m 0 (mod ) and m (mod ) with p an odd prime. Then Rank E m,p (Q). Before we proceed ahead for the proof of above stated two theorems, in the next section we develop necessary background material needed for better exposition and clarity of presentation in this paper.. Preliminaries: notations, definitions and related known results In reference to the notional convention through out this article we denote the families of elliptic curvesy = x m x+p bye m,p. For the convenience of the reader, we describe some related material from [] that includes some basic concepts of elliptic curves over rational field Q and well known results. An elliptic curve over the field Q of rational numbers is a curve E defined by a Weierstrass equation where a,b Z, and E : y = x +ax+b, = (a +b ) 0. In other words, the above condition is equivalent to the cubic equation x +ax+b = 0, having three distinct complex roots. In essence, an elliptic curve E, can be thought of a curve in projective space P, with homogeneous equation y z = x +axz +bz and a point, namely [0,,0], at infinity which we denote by. Note that all the vertical lines meet at the point. Let E(Q) := {(x,y) Q : y = x +ax+b} { }. One can define the law of addition for the points on the elliptic curve which, in literature, is known as chord-tangent law of addition (for details, see []). The set of all points of an elliptic curve E forms an abelian group, of which E(Q) is a subgroup. In particular, the point at is the identity element and inverse of A = (x, y) E(Q) is A := (x, y). Fundamental Mordell Theorem [] says that the group E(Q) of all rational points of an elliptic curve E is a finitely generated abelian group, which means that E(Q) = Z r +TorsE(Q),

3 ON THE FAMILY OF ELLIPTIC CURVES y = x m x+p where r is an uniquely determined non-negative integer called the rank of elliptic curve and TorsE(Q) is the finite abelian group consisting of all elements of finite order in E(Q). The torsion subgroup of E(Q) is well-understood". Nagell-Lutz Theorem([], [0]) and Mazur Theorem [] gave complete classification of a torsion subgroup of an elliptic curve over the rational field. The notion of the rank of an elliptic curve has been studied. However, its characterization is, in general, a difficult task. A primary reason of this difficulty is that the rank can not be obtained effectively from the coefficients a,b of the curve s equation. Now in search of either an exact rank of E(Q) or a lower bound on rank of E(Q), in literature, plenty of computational ways have been developed, many of them being either computationally complex or exploit heavy mathematical machinery.. Torsion Subgroup of E m,p In this section, our main aim is to prove Theorem.. A technique that we use here is to reduce E m,p over finite fields. Let E : y = x +ax+b be an elliptic curve, where a,b Z. For given prime p, E is treated as a curve over the finite field F p, where a,b,x,y F p. Let (E) = (a +b ) be the discriminant of elliptic curve E. Now if p (E), then the roots of cubic equation x + ax + b are all distinct and in that case E becomes an elliptic curve over F p. Essentially this is an instance when one says that at p, E has a good reduction and E(F p ) is known as the group of F p -points of E. Now given good reduction of E at p, application of Theorem. gives an injective map from the group of rational torsion points TorsE(Q) into the group E(F p ). The following theorem and lemmas will be the main ingredients for proving the Theorem.. Theorem. ([], Theorem.). The restriction of the reduction homomorphismr p TorsE(Q) : TorsE(Q) E p (F p ) is injective for any odd prime p, where E has good reduction and r TorsE(Q) : TorsE(Q) E (F ) has kernel at most Z/Z when E has good reduction at. Lemma.. There is no point of order two in E m,p (Q) for every positive integer m and every prime p. Proof. Let A = (x,y) be a point in E m,p (Q) of order two. Then A = 0 A = A y = 0 and x 0. Thus x m x+p = 0. Since A has finite order, x must be an integer (by Nagell-Lutz Theorem). Therefore the above equation gives, m = x + p x, which implies that x {±,±p,±p }. Hence m {±p,p ±p,p ±}, which contradicts that m is an integer. By doubling a point on elliptic curve, we mean adding the point to itself (see []). On E m,p, if P = (x,y) is any point, then by the addition law, we get the doubling of P, denoted

4 ABHISHEK JUYAL, SHIV DATT KUMAR P = (x,y ), is given by and x = x +m x +m xp, () (x m x+p ) y = y x m (x x). () y Lemma.. E m,p (Q) does not contain a point of order for any positive integer m with m 0 (mod ) and for any prime p. Proof. Assume contrary that there exists a point A = (x,y) E m,p (Q) such that A = O A = A x-coordinate of A = x-coordinate of A. Therefore x +x m +m xp = x, (x m x+p ) equivalently x m x +p x m = 0. () If m 0 (mod ), then under modulo, equation () has no solution, which implies that equation () has no rational solution. Thus E m,p (Q) has no point of order for m 0 (mod ) and for any prime p. Lemma.. E m,p (Q) does not contain a point of order for any positive integer m with m 0 (mod ), and for any prime p with gcd(m,p) =. Proof. Initial part of the proof of this lemma is quite similar to the proof of Lemma. till equation (). From equation () it is clear that x m. Now here two cases arises: Case. When x 0 (mod ). Suppose the prime factorization of x contains a and the prime factorization of m contains b then it is clear that b a. Now, write equation () as a+ (a ) b+a+ (a )+ a+ (a ) b (a ) = 0. () Where a,a,a and a are some integers. Under modulo (a+), () has no solution which implies that () too has no rational solution. Case. When x 0 (mod ). Then equation () can be written as which is quadratic in m with discriminant Equation () also gives (m ) +(x )m (x +p x) = 0, () = x(x +p ). m = x +,

5 ON THE FAMILY OF ELLIPTIC CURVES y = x m x+p a further simplification leads to = (m +x ). By comparing both values of, we get (m +x ) = x(x +p ). () Equation () implies there existsn N such thatx(x +p ) = n. Now gcd(x,x +p ) = or p or p. If gcd(x,x +p ) = p or p, then from () p m which contradicts that gcd(m,p) =. Therefore, gcd(x,x +p ) = and we can factor right hand side of equation () as:.(a) { x = α, x +p = β,.(a) can be rewritten as or.(b) { x = α, (α ) +p = β. (α ) +p = β, () Since equation () has no solution under modulo, hence equation () too has no rational solution. Given that x 0 (mod ), case (b) is impossible. Hence we conclude that E m,p (Q) does not have a point of order for m 0 (mod ), and for any prime p with gcd(m,p) =. We are now equipped with much of needed machinery and ready to prove the Theorem.. Proof of Theorem.. We split the proof in two cases which together complete the proof of this theorem. Case. When p. In this case = ( m +p ), hence E m,p has good reduction at. Reducing E m,p over F, results in following: (a). If p (mod ). Depending upon whether m 0, or (mod ), E m,p reduces to y = x +,y = x x+,y = x x+ respectively and the corresponding cardinality of E m,p (F ) being,,. (b). If p (mod ). E m,p reduces to y = x +,y = x x +,y = x x + according to m 0, or (mod ) respectively with corresponding cardinality of E m,p (F ) being,,. An application of Theorem. and Lagrange theorem, furnishes the complete list of possible orders of TorsE m,p (Q) as,,,, and only. Lemma., Lemma., Lemma. guarantee that the curve E m,p does not have points of order and. Thus we conclude that TorsE m,p (Q) = {O}. Case. When p =. Note that and, hence E m, : y = x m x+ has good reductions at and. Reducing E m,p over F, and F we have: (c). Over F, we get E m, : y = x +, or y = x x +, depending upon whether m 0 or, (mod ) and in that case corresponding cardinality of E m, (F ) being and respectively. Thus in this case we have TorsE m, (Q) =,, or of which the only

6 ABHISHEK JUYAL, SHIV DATT KUMAR probable values are and because of the Lemma.. (d). Over F we get, y = x +, y = x x+, y = x x+ or, y = x x+ according as m 0,,,or (mod ) respectively, with the cardinality of E m, (F ) being,0,0,0 respectively. Thus the possible values for the cardinality of TorsE m,p (Q) are,,, or 0 of which, and 0 are again not possible by the same reasoning as in previous cases. Thus TorsE m,p (Q) = or,. These two subcases (c) and (d) together imply that and cannot be the cardinality of TorsE m,p (Q) because in the former case all the non-trivial points must have order, whereas the same in latter case would be, leading to a contradiction. We thus conclude that TorsE m,p (Q) = {O}. Remark. The condition gcd(m, p) = has been imposed in the proof of Theorem (.) because we have an example of an elliptic curve, namely y = x x +, which has its torsion subgroup isomorphic to Z. However, we did not find any other example for which the torsion subgroup is non-trivial.. The Rank of E m,p In this section we produce a subfamily of minimum rank. To achieve this, it is sufficient to find at least two independent points, say A m,p and B m,p, on each curve in E m,p. First we note that, the point B m,p = (m,p) lies on every curve E m,p and the x-coordinate of B m,p is (m mp )/p. Thus if gcd(m,p) =, then the order of B m,p is infinite, and hence we get a subfamily of E m,p of rank at least one. To get a subfamily of minimum rank two we will use the following result. Theorem. ([], p. ). Let E(Q) (resp. E(Q)) be the group of rational points (resp., double of rational points) on an elliptic curve E, and suppose E has trivial rational torsion. Then the quotient group E(Q)/E(Q) is an elementary abelian -group of order r, where r is the rank of E(Q). On each curve in E m,p one can find the obvious points namely, A m,p = (0,p),B m,p = (m,p). We will now construct a subfamily of E m,p for which the above two points will be linearly independent. We first prove few results which will help us achieving this. Lemma.. Let A = (x,y ) and B = (x,y) be points in E m,p (Q), such that A = B and x Z. Then () x Z, () x m (mod ). Proof. Substituting x = u, gcd(u,s) = in () and after simplifying we get, s (m p x )s +(m x p )us +m u s x u s+u = 0 ()

7 ON THE FAMILY OF ELLIPTIC CURVES y = x m x+p From (), it is clear that s u, so s {,}, and hence x Z. Again, we can write equation () as (x +m ) = (x (x m x+p )+xp ), which implies that (x +m ), i.e., x m (mod ). Lemma.. The point A m,p = (0,p) is an element of E m,p (Q)\E m,p (Q) for any positive integer m with m (mod ) and any odd prime p. Proof. Suppose A m,p = C for some C = (x,y) E m,p (Q). Then we have x +m x +m xp = 0, (x m x+p ) or x +m x +m xp = 0. () The simplification of above equation leads to (x +m ) = xp. (0) From equation (0) we observe that x = k for some k Z, i.e., x is an even number. Substituting the value of x in equation (), we get k +m k +m k p = 0. () It follows that for any integer k, under modulo, equation () has no solution. Consequently equation () has no rational solution. Therefore, A m,p / E m,p. Lemma.. The point B m,p = (m,p) is an element of E m,p (Q)\E m,p (Q) for any positive integer m with m (mod ) and for any prime p. Proof. Suppose B m,p = (m,p) = C for some C = (x,y) E m,p (Q). Then, Simplifying it, we get which can be rewritten as x +m x +m xp (x m x+p ) = m. x mx +m x +(m p )x+m p m = 0, (x m) (x m) m (x m)p p m+m = 0. () Taking lemma. into consideration, we substitute x m = s, which results in simplification of () as: (s m ) = (s+m)p. The above equation holds only if (s+m) = w, for some w Z. Since m (mod ), s + m (mod ) leading to a contradiction that it is a perfect square. This completes the proof. Lemma.. The point A m,p +B m,p = ( m,p) is an element of E m,p (Q)\E m,p (Q) for any positive integer m with m (mod ) and for any prime p.

8 ABHISHEK JUYAL, SHIV DATT KUMAR Proof. With arguments similar to the Lemma (.), we can say that A m,p +B m,p = ( m,p) E m,p(q) for m (mod ) and for any prime p. We are now ready to prove Theorem.. Proof of Theorem.. To show that RankE m,p (Q), we first claim that H = {[O], [A m,p ], [B m,p ],[A m,p ] + [B m,p ]}, is a subgroup of E m,p /E m,p of order, for the values of m and p satisfying the conditions of Theorem.. We have to show that the elements of H are all distinct. To this end we first observe that [A m,p ] [O], [B m,p ] [O] and [A m,p +B m,p ] [O]. Now suppose [A m,p ] = [B m,p ]. Then [A m,p +B m,p ] = [A m,p ]+[B m,p ] = [A m,p ] = [O], which is a contradiction. Similarly it can be shown that [A m,p ] [A m,p + B m,p ] and [B m,p ] [A m,p +B m,p ]. Hence H is a subgroup of E m,p /E m,p of order. Secondly we claim thata m,p andb m,p are linearly independent points ine m,p (Q). Suppose on the contrary that there exist integers u and v such that ua m,p + vb m,p = O. Without loss of generality, we may further assume that u is the smallest positive integer with this property. Now we have the following cases: () If u is even and v is odd, then [O] = [ua m,p + vb m,p ] = u[a m,p ] + v[b m,p ] or [O] = [B m,p ], which contradicts Lemma.. () If u is odd and v is even, then [O] = [ua m,p + vb m,p ] = u[a m,p ] + v[b m,p ] or [O] = [A m,p ], again contrary to what Lemma. asserts. () If u and v are both odd, then [O] = [A m,p + B m,p ], and this contradicts to Lemma.. () Ifuandv are even, thenu = u, v = v for someu,v. We have[u A m,p +v B m,p ] = [O]. This implies that u A m,p +v B m,p is a rational point of order. But E m,p (Q) has only trivial torsion point. Therefore, u A m,p +v B m,p = O, but then this contradicts the minimality of u. We have thus shown that A m,p and B m,p are linearly independent points in E m,p (Q). Now by Theorem. the cardinality of E m,p /E m,p, is r, where r is the rank of E m,p (Q). By our first claim, E m,p (Q) has at least points which means that the rank r, of E m,p (Q) is at least for any positive integer m, with m 0 (mod ) and m (mod ) and p an odd prime. This concludes the theorem.. Examples of curves with high rank Over the rational field Q, one typical way to find elliptic curves with high rank is to construct the families of elliptic curves with high generic rank and thereafter, we search for an adequate specialization together with effective sieving tools. The Mestre-Nagao sum is used very often, ([], []). Over Q, let E be an elliptic curve, and p be a prime, set a p = a p (E) = p+ E(F p ). Given a fixed integer N, Mestre-Nagao sum is defined as: S(N,E) = = p N,p prime p N,p prime ( p E(F p ) a p + p+ a p log(p). ) log(p)

9 ON THE FAMILY OF ELLIPTIC CURVES y = x m x+p There do exist experimental verification that one generally expect that those are the larger values of S(N,E) to which high rank curves correspond. In this hope, we searched for curves E = E m,p in the range m 0000; p 0000, for which S(,E) >,S(,E) >. After this initial sieving, we calculated the rank of the remaining curves with sage []. Table comprehensively summarizes the results. Table. rank [m, p] [, ][, ],[, ],[, 00],[, ],[, 0],[, ],[, ], [, ],[0, ],[0, ],[, 0],[, ],[, 0],[, ], [, ],[, ],[, 0],[, ],[, 0],[, ],[, ], [, ],[0, ],[0, ],[, ],[, ],[, ],[, ], [, ],[, ],[, ],[, 0],[, ],[, ],[, ], [, ],[, 0],[, ],[, ],[, ],[, ],[, ], [, ],[, 0],[, 0],[, ],[, ],[, ],[, 0], [, ],[, ],[, ],[, ],[, ],[, ],[0, ], [, 0],[, ],[0, 0],[, 0],[, ],[, ], [0, ],[, ],[, ],[, ], [, ],[, ],[, ],[, ],[, ],[, ],[, 0], [, ],[, ],[, ],[, ],[, ],[, ],[, ], [, ],[0, 0],[, ],[0, ],[, ],[, ], [, ],[, ],[, ],[, ],[, ],[, ], [, ],[, ],[, ],[, 0],[, ],[, ], [, ],[, ],[, ],[, 0],[, ],[, ], [, ],[, ],[, ],[, 00],[0, ],[0, ], [, ],[, ],[, ],[0, ],[, ],[, ], [, 00],[, ],[, ],[, ],[, ],[, ], [, ],[, 0],[, ],[, 0],[, ],[, ], [, ],[, ],[, ] [, ],[, ],[, ],[, ],[, ],[, 0], [, ],[, 0],[, ],[, ],[, ],[, ], [, ],[, ],[, ],[, ],[, ],[, ], [, ],[, ],[, ] Acknowledgment: The authors are grateful to the reviewer for constructive feedback which has helped in improving the expositions and technical quality of the manuscript. The first author sincerely thanks the Harish-Chandra Research Institute, Allahabad for providing research facilities to pursue his research work and is indebted to Prof. Kalyan Chakraborty (HRI Allahabad) for his continuous support and encouragement.

10 0 ABHISHEK JUYAL, SHIV DATT KUMAR References [] A. Antoniewicz, On a Family of Elliptic curves, Universitatis Iagellonicae Acta Mathematica : (00),. [] Barry Mazur, Modular curves and the Eisenstein ideal, Publications Mathématiques I.H.E.S. (),. [] D. Husemöller, Elliptic Curves, Springer-Verlag, New York (). [] E. Brown, B. T. Myers, Elliptic curves from Mordell to Diophantus and Back, Amer. Math. Monthly, 0: (00),. [] E.V.Eikenberg, Rational points on some families of elliptic curves, Ph.D. thesis, University of Maryland, 00. [] J. E. Cremona, Algorithms for modular Elliptic Curves, nd ed., Cambridge University Press, New York (). [] J.F. Mestre, Construction de courbes elliptiques sur Q de rang, C. R. Acad. Sci. Paris Ser. I, (),. [] J. H. Silverman and J. Tate, Rational Points On Elliptic Curves, Undergraduates Texts in Mathematics, Springer-Verlag, New York (). [] K. Nagao, An example of elliptic curve over Q with rank 0, Proc. Japan Acad. Ser. A Math. Sci. (),. [0] Lutz, E., Sur l equation y = x Ax B dans les corps p-adic. J. Reine Angew. Math. (), -. [] Mordell, L. J., On the rational solutions of the indeterminate equations of the third and fourth degrees. Procd. Camb. Philos. Soc. (), -. [] Nagell, T., Solution de quelque problémes dans la théorie arithmétique des cubiques planes du premier genre. Wid. Akad. Skrifter Oslo I (), Nr.. [] P. Tadić, The rank of certain subfamilies of the elliptic curve Y = X X+T, Annales Mathematicae et Informaticae, (0), -. [] P. Tadić, On the family of elliptic curve Y = X T X +, Glasnik Matematicki, () (0), -. [] Sage software, Version.. Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad - 00, India address, Abhishek Juyal: abhinfo@gmail.com address, Shiv Datt Kumar: sdt@mnnit.ac.in