A quantitative primitive divisor result for points on elliptic curves

Ingram, Patrick

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 609-634.

Soient $E/K$ une courbe elliptique définie sur un corps de nombres et $P\in E\left(K\right)$ un point d’ordre infini. Il est naturel de se demander combien de nombres entiers $n\ge 1$ n’apparaissent pas comme ordre du point $P$ modulo un idéal premier de $K$. Dans le cas où $K=\mathbb{Q}$, $E$ une tordue quadratique de $y^2=x^3-x$ et $P\in E\left(\mathbb{Q}\right)$ comme ci-dessus, nous démontrons qu’il existe au plus un tel $n\ge 3$.

Let $E/K$ be an elliptic curve defined over a number field, and let $P\in E\left(K\right)$ be a point of infinite order. It is natural to ask how many integers $n\ge 1$ fail to occur as the order of $P$ modulo a prime of $K$. For $K=\mathbb{Q}$, $E$ a quadratic twist of $y^2=x^3-x$, and $P\in E\left(\mathbb{Q}\right)$ as above, we show that there is at most one such $n\ge 3$.

[1] A. S. Bang, Taltheoretiske Undersølgelser. Tidskrift f. Math. 5 (1886).

[2] Y. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539 (2001), (with an appendix by M. Mignotte). | MR | Zbl

[3] A. Bremner, J. H. Silverman and N. Tzanakis, Integral points in arithmetic progression on $y^2=x(x^2-n^2)$. J. Number Theory, 80 (2000). | MR | Zbl

[4] Y. Bugeaud, P. Corvaja, and U. Zannier,An upper bound for the G.C.D. of $a^n-1$ and $b^n-1$. Mathematische Zeitschrift 243 (2003). | MR | Zbl

[5] R. D. Carmichael,On the numerical factors of the arithmetic forms ${\alpha }^n\pm {\beta }^n$. Annals of Math. 2nd series, 15 (1914), 30–48 and 49–70. | JFM

[6] G. Cornelissen and K. Zahidi, Elliptic divisibility sequences and undecidable problems about rational points. J. Reine Angew. Math. 613 (2007). | MR | Zbl

[7] S. David, Minorations de formes linéaires de logarithmes elliptiques. Mém. Soc. Math. France No. 62 (1995). | EuDML | Numdam | MR | Zbl

[8] G. Everest, G. McLaren, and T. Ward, Primitive divisors of elliptic divisibility sequences. J. Number Theory 118 (2006). | MR | Zbl

[9] P. Ingram, Elliptic divisibility sequences over certain curves. J. Number Theory 123 (2007). | MR | Zbl

[10] P. Ingram, Multiples of integral points on elliptic curves. J. Number Theory, to appear (arXiv:0802.2651v1) | MR

[11] P. Ingram and J. H. Silverman, Uniform bounds for primitive divisors in elliptic divisibility sequences. (preprint)

[12] K. Ireland and M. Rosen. A Classical Introduction to Modern Number Theory. Volume 84 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1998. | MR | Zbl

[13] PARI/GP, version 2.3.0, Bordeaux, 2005, http://pari.math.u-bordeaux.fr/.

[14] B. Poonen, Characterizing integers among rational numbers with a universal-existential formula. (arXiv:math/0703907) | MR

[15] K. F. Roth, Rational approximations to algebraic numbers. Mathematika 2 (1955). | MR | Zbl

[16] A. Schinzel, Primitive divisors of the expression $A^n-B^n$ in algebraic number fields. J. Reine Angew. Math. 268/269 (1974). | MR | Zbl

[17] R. Shipsey, Elliptic divisibility sequences. Ph.D. thesis, Goldsmiths, University of London, 2001.

[18] J. H. Silverman, The arithmetic of elliptic curves. Volume 106 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1986. | MR | Zbl

[19] J. H. Silverman, Wieferich’s criterion and the $abc$-conjecture. J. Number Theory 30 (1988). | MR | Zbl

[20] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. Volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. | MR | Zbl

[21] J. H. Silverman, Generalized greatest common divisors, divisibility sequences, and Vojta’s conjecture for blowups. Monatshefte für Mathematik 145 (2005). | MR

[22] C. L. Stewart, Primitive divisors of Lucas and Lehmer numbers. In Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976), Academic Press, London, 1977. | MR | Zbl

[23] R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. Acta Arithmetica 67 (1994). | MR | Zbl

[24] P. Vojta, Diophantine approximations and value distribution theory. Volume 1239 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1987 | MR | Zbl

[25] M. Ward, Memoir on elliptic divisibility sequences. Amer. J. Math. 70 (1948). | MR | Zbl

[26] K. Zsigmondy, Zur Theorie der Potenzreste. Monatsh. Math. 3 (1892).