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Some Families of Mordell Curves Associated to Cubic Fields

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Some Families of Mordell Curves Associated to Cubic Fields View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 160 (2003) 217–231 www.elsevier.com/locate/cam Some families of Mordell curves associated to cubic ÿelds Katsuya Miyake1 Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan Received 7 October 2002; received in revised form 3 August 2003 Abstract We introduce some Mordell curves of two di-erent natures both of which are associated to cubic ÿelds. One set of them consists of those elliptic curves whose rational points over the rational number ÿeld are described by or closely related to cubic ÿelds. The other is a one-parameter family of Mordell curves which gives all (cyclic) cubic twists and all quadratic twists of the Fermat curve X 3 + Y 3 + Z3 =0. c 2003 Elsevier B.V. All rights reserved. Keywords: Mordell curves; Cubic ÿelds Introduction 1. One of the deepest and most interesting problem on arithmetic of elliptic curves over the rational number ÿeld is certainly the Birch and Swinnerton-Dyer conjecture [1]. It claims that certain basic analytic invariants of the Hasse–Weil L-function of an elliptic curve would be expressed with the periods and some algebraic invariants. A reÿned description of the conjecture looks quite analogous to the residue of the Dedekind zeta-function of an algebraic number ÿeld at the pole at s=1 although, at the present, we cannot see any mathematical reasons for the resemblance. It has been proved by the epoch-making work [15] of Wiles and the successive [2] by Breuil et al. that every elliptic curve over the rational number ÿeld Q is modular and hence that its Hasse-Weil L-function L(s) coincides with the L-function of a related modular form. Therefore, L(s) has a functional equation and is analytically continued all over the complex plane as a meromorphic This article was prepared for a talk in International Conference on Special Functions and their Applications, September 23–27, 2002, at The Institute of Mathematical Sciences, Chennai, India. 1 The author was partly supported by the Grant-in-Aid for Scientiÿc Research (C) (2) No. 14540037, Japan Society for the Promotion of Science while he prepared this work. E-mail address: [email protected] (K. Miyake). 0377-0427/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-0427(03)00624-1 218 K. Miyake / Journal of Computational and Applied Mathematics 160 (2003) 217–231 −r function. Its order r of zero at s=1 and the limit value lims→1(s−1) L(s) is hence well deÿned. The order r is called the analytic rank of the elliptic curve, and conjectured to coincide with the Mordell– Weil rank of the curve, that is, the Z-rank of the free part of the Z-module of the Q-rational points. (This is the easy part to state of the Birch and Swinnerton-Dyer conjecture. The precise description of the conjecture is, for example, found in the textbook [13] of Silverman.) It should also be noted that the L-functions of two elliptic curves over Q coincide with each other if and only if the curves are isogenous over Q. This is now an immediate conclusion of the isogeny theorem (for Q) of the celebrated work [4] of Faltings in which he proved the theorem for every algebraic number ÿeld along the way to show the Mordell conjecture. Well, the analytic basis of the Birch and Swinnerton-Dyer conjecture has thus been settled. It is, however, far out of my task here to show any substantial materials on the conjecture. 2. The purpose of my talk is to introduce four subfamilies of Mordell curves which are associated with quadratic and cubic ÿelds which may add some amusing sceneries to arithmetic of elliptic curves, I hope. The curves are 2 3 4 M1(c):y = x − 2 c(27c +4); 2 3 4 3 M2(c):y = x +2 3 c(27c +4); 2 3 4 3 3 M3(c):y = x +2 3 c (27c +4); 2 3 4 3 3 3 M4(c):y = x +2 3 c (27c +4) ; where c ∈ Q −{0; − 4 }. 27 In 1914 Mordell investigated integral solutions of the equation 2 3 M0(k):y = x + k with k ∈ Z in his article [9]. Then Nagel [11] studied rational points on the curve when k was a 3 2 perfect cube. In his paper [5] Fueter worked on the equation√ by expressing it as x = y + D and factorizing the right-hand side in the quadratic ÿeld Q( −D). The papers [3] of Cassels and [12] of Selmer should be classic references. In 1998 Gebel, Peth˝o and Zimmer published their article [6] with the title “On Mordell’s Equation” after which I chose the terminology ‘Mordell curves’ in my present talk. The ÿrst family of Mordell curves M1(c) is related with the cubic ÿeld Kc = Q() where is a ÿxed root of the cubic equation P(c; X ) = 0 where P(c; X )=X 3 + X 2 + c; this is reducible over Q if and only if c = −r2(r + 1) with r ∈ Q; here let us assume that P(c; X ) is irreducible over Q. Then we are able to describe the set of Q-rational points of the curve M1(c) by a certain subset of the cubic ÿeld Kc (see Proposition 1 in Section 3 and Theorem 1 in Section 4). It should be noted that every cubic ÿeld is generated by a root of P(c; X ) = 0 for some c ∈ Q (see Proposition 2). The Mordell–Weil rank of M1(c) is at least equal to 1 with one exception for c = − 2 ; (see Proposition 1). The curves M (c) and M (c) will appear in connection with the ÿrst 27 2 3 (see Sections 6 and 8). See Mestre [7,8] for Mordell curves with large Mordell–Weil ranks. K. Miyake / Journal of Computational and Applied Mathematics 160 (2003) 217–231 219 The last one of our four curves comes out of a quite di-erent way from the other three. It is a twist of the Fermat cubic curve X 3 + Y 3 + Z3 = 0. As it is well known, the curve 3 3 3 F0(A):X + Y = AZ with A ∈ Q −{0} is transformed to a Mordell curve 2 3 4 3 2 M5(A):y = x − 2 3 A over Q; e.g. cf. [3] (see Remark at the end of Section 11). If we put A = −1, then we have the 2 3 4 3 short form y = x − 2√3 of the Fermat cubic curve. Therefore M5(A) is a twist of M5(−1) over 3 2 3 4 3 the pure cubic ÿeld Q√( A). It is easy to see that the square twist dy = x − 2 3 of M5(−1) over the quadratic ÿeld Q( d) is given by the Mordell curve 2 3 4 3 3 M6(d):y = x − 2 3 d : Hence our family M4(c) is a subfamily of M6(d) for d = −c(27c + 4). (Note that this d is the discriminant of the cubic polynomial P(c; X ).) We obtain our M4(c) by a quite di-erent way from 3 3 3 this; it comes out√ as a twist of X + Y + Z = 0 over the splitting ÿeld of P(c; X ) over Q which is equal to Kc( d) in general; (see Proposition 6 in Section 9). If c = −r2(r + 1) with r ∈ Q −{0; −1; − 2 ; 1 } then we have a rational root = r of the 3 3 equation P(c; X ) = 0; hence Kc = Q, and the splitting ÿeld coincides with√ the quadratic ÿeld Q( (r + 1)(1 − 3r)); (see Proposition 7 in Section 9). This is equal to Q( )ifweput = (1 − 3r)=(r + 1). Our family√ M4(c), therefore, contains a quadratic twist of the Fermat cubic curve X 3 + Y 3 + Z3 = 0 over Q( ) for c = −r2(r + 1) and r =(1− )=(3 + ). Part I. Mordell curves associated to cubic ÿelds 3. To be more precise let us start with the elliptic curve 3 3 2 E1(c):w = P(c; u)=u + u + c: This combines the rational points of M1(c) with elements of the cubic ÿeld Kc when the cubic polynomial P(c; X ) is irreducible over Q. The essential part of the following proposition ((i) and the former half of (ii)) is due to Toshiharu Kishi. I also owe some information and computation on Mordell curves to Masanari Kida and Yasuhiro Kishi. Hereafter in this talk we assume that the parameter c is not equal to either 0 or − 4 . Hence the 27 equation P(c; X ) = 0 does not have any double roots. Proposition 1. (i) The short form of E (c) for c ∈ Q −{0; − 4 } is M (c). Therefore, E (c) is an 1 27 1 1 elliptic curve with a rational point at inÿnity. (ii) M1(c) has a rational point 2 4 2 4 3 6 3 4 4 2 (x0;y0)=(2 3 c +2 3c +4=9; 2 3 c +2 3 c +60c − 8=27): This is not a torsion point unless c = − 2 , and hence its Mordell–Weil rank is greater than or 27 equal to 1 if c = − 2 . 27 220 K. Miyake / Journal of Computational and Applied Mathematics 160 (2003) 217–231 (iii) The exceptional curve M (− 2 ):y2 = x3 +(4 )3 has M–W rank =0.
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