On a class of elliptic curves with rank at most two. (English) Zbl 0833.11022
The author considers the well-known elliptic curve given by the Weierstraß equation \(y^2= x^3+ px\) for a given prime \(p\). For \(p\equiv 5\pmod 8\) the corresponding curves were investigated by A. Bremner and J. W. S. Cassels [Math. Comput. 42, 247-264 (1984; Zbl 0531.10014)]and A. Bremner [Number theory and applications, NATO ASI Ser., Ser. C 265, 3-22 (1989; Zbl 0689.14010)]and their computational efforts support the conjecture that the rank of the Mordell-Weil group should be 1 for these curves.
The present paper deals with those curves for which \(p\equiv 1\pmod 8\). It is known that the rank can only have the values 0, 1, or 2 for these primes. These author has gathered much evidence to support his claim that the rank cannot equal 1, provided it is positive and \(p\) can be expressed as \(p= a^2+ 64b^2\). In fact he has explicitly computed the generators of the group of rational points and the Mordell-Weil lattice invariant \(\tau\) for each prime \(p\) with \(p< 50000\) of the said form.
The present paper deals with those curves for which \(p\equiv 1\pmod 8\). It is known that the rank can only have the values 0, 1, or 2 for these primes. These author has gathered much evidence to support his claim that the rank cannot equal 1, provided it is positive and \(p\) can be expressed as \(p= a^2+ 64b^2\). In fact he has explicitly computed the generators of the group of rational points and the Mordell-Weil lattice invariant \(\tau\) for each prime \(p\) with \(p< 50000\) of the said form.
Reviewer: R.J.Stroeker (Rotterdam)
MSC:
| 11G05 | Elliptic curves over global fields |
| 14H52 | Elliptic curves |
| 11D25 | Cubic and quartic Diophantine equations |
| 11-04 | Software, source code, etc. for problems pertaining to number theory |
| 14Q05 | Computational aspects of algebraic curves |
Online Encyclopedia of Integer Sequences:
Primes p == 1 (mod 8), p = a^2 +64*b^2 such that y^2 = x^3 + p*x has rank 0.Primes p == 1 (mod 8), p = a^2 + 64*b^2 such that y^2 = x^3 + p*x has rank 2.
References:
| [1] | A. Bremner and J. W. S. Cassels, On the equation \?²=\?(\?²+\?), Math. Comp. 42 (1984), no. 165, 257 – 264. |
| [2] | A. Bremner, On the equation \( {Y^2} = X({X^2} + p)\), Number Theory and Applications , Kluwer, Dordrecht, 1989, pp. 3-23. · Zbl 0689.14010 |
| [3] | Joe P. Buhler, Benedict H. Gross, and Don B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp. 44 (1985), no. 170, 473 – 481. · Zbl 0606.14021 |
| [4] | Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. · Zbl 0786.11071 |
| [5] | L. J. Mordell, The diophantine equation \?\(^{4}\)+\?\?\(^{4}\)=\?²., Quart. J. Math. Oxford Ser. (2) 18 (1967), 1 – 6. · Zbl 0154.29702 · doi:10.1093/qmath/18.1.1 |
| [6] | H. E. Rose, On a class of elliptic curves with rank at most two, Math. Comp. 64 (1995), no. 211, 1251 – 1265, S27 – S34. · Zbl 0833.11022 |
| [7] | Karl Rubin, The one-variable main conjecture for elliptic curves with complex multiplication, \?-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 353 – 371. · Zbl 0741.11028 · doi:10.1017/CBO9780511526053.015 |
| [8] | Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
