Integral points on elliptic curves defined by simplest cubic fields. (English) Zbl 0983.11031

Duquesne, Sylvain

Exp. Math. 10, No. 1, 91-102 (2001).

For an integer \(m\) let \(f_m(X)=X^3 + mX^2 -(m+2)X +1\). If \(m^2+3m+9\) is square-free, then the cubic field defined by \(f_m(X)\) was called a simplest cubic field by D. Shanks [Math. Comput. 28, 1137-1152 (1974; Zbl 0307.12005)]. In the present paper the integral points lying on the elliptic curve \(E_m\) are studied, where \(E_m\) is defined by the equation \(Y^2 = f_m(X)\). Using the method developed by R. J. Stroeker and N. Tzanakis [Acta Arith. 67, 177-196 (1994; Zbl 0805.11026)], by J. Gebel, the reviewer and H. G. Zimmer [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)] and by N. Smart [Math. Proc. Camb. Philos. Soc. 116, 391-399 (1994; Zbl 0817.11031)], all integral points lying on \(E_m\) are computed for \(1 \leq m \leq 1000\). From the experimental result, several observation are deduced. Some of them are proved in the second half of the paper.
The first general result is that if \(m\) is even, then there are no integral points on \(E_m\) with positive \(x\) coordinate. The point \([0,1]\) has infinite order in \(E_m({\mathbb Q})\). The most interesting result is that if \(f_m(X)\) generates a simplest cubic field, then \([0,1]\) is a generator of \(E_m({\mathbb Q})\). Moreover there are no other integral points on \(E_m\) that are positive multiples of \([0,1]\), apart from \(2[0,1]\) when \(m\) is odd. This means a complete characterization of integral points of \(E_m({\mathbb Q})\) whenever its rank is one. Infinite subfamilies of rank three and five are also given.

Reviewer: Attila Pethő (Debrecen)

MSC:

11G05	Elliptic curves over global fields
11Y50	Computer solution of Diophantine equations
11D25	Cubic and quartic Diophantine equations
14H52	Elliptic curves

Keywords:

elliptic diophantine equation; elliptic curve; simplest cubic field; integral points

Citations:

Zbl 0805.11026; Zbl 0816.11019; Zbl 0817.11031; Zbl 0307.12005

Software:

mwrank

References:

[1]	DOI: 10.1093/qmath/20.1.129 · Zbl 0177.06802 · doi:10.1093/qmath/20.1.129
[2]	Cohen H., A course in computational algebraic number theory (1993) · Zbl 0786.11071 · doi:10.1007/978-3-662-02945-9
[3]	Cremona J. E., ”mwrank, a program for 2-descent on elliptic curves over Q” (1998)
[4]	David S., Minorations de formes linéaires de logarithmes elliptiques (1995)
[5]	Gebel J., Acta Arith. 68 (2) pp 171– (1994)
[6]	Husemoller D., Elliptic curves (1987) · Zbl 0605.14032
[7]	Lang S., Elliptic curves: Diophantine analysis (1978) · Zbl 0388.10001
[8]	DOI: 10.1090/S0025-5718-98-00958-2 · Zbl 0920.11074 · doi:10.1090/S0025-5718-98-00958-2
[9]	Pethö A., Math. Comp. 57 (196) pp 777– (1991)
[10]	DOI: 10.1090/S0025-5718-1974-0352049-8 · doi:10.1090/S0025-5718-1974-0352049-8
[11]	Shioda T., Comment. Math. Univ. St. Paul. 39 (2) pp 211– (1990)
[12]	DOI: 10.1090/S0025-5718-1988-0942161-4 · doi:10.1090/S0025-5718-1988-0942161-4
[13]	DOI: 10.1090/S0025-5718-1990-1035944-5 · doi:10.1090/S0025-5718-1990-1035944-5
[14]	DOI: 10.1017/S0305004100072698 · Zbl 0817.11031 · doi:10.1017/S0305004100072698
[15]	Smart N. P., The algorithmic resolution of Diophantine equations (1998) · Zbl 0907.11001 · doi:10.1017/CBO9781107359994
[16]	Stroeker R. J., Acta Arith. 67 (2) pp 177– (1994)
[17]	DOI: 10.1090/S0025-5718-1987-0866122-8 · doi:10.1090/S0025-5718-1987-0866122-8
[18]	DOI: 10.1090/S0025-5718-1987-0866125-3 · doi:10.1090/S0025-5718-1987-0866125-3