 A quantitative primitive divisor result for points on elliptic curves
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 $K=ℚ$, $E$ une tordue quadratique de ${y}^{2}={x}^{3}-x$ et $P\in E\left(ℚ\right)$ 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 $K=ℚ$, $E$ a quadratic twist of ${y}^{2}={x}^{3}-x$, and $P\in E\left(ℚ\right)$ as above, we show that there is at most one such $n\ge 3$.

Reçu le : 2008-05-01
Publié le : 2010-03-22
DOI : https://doi.org/10.5802/jtnb.691
Classification : 11G05,  11B39
@article{JTNB_2009__21_3_609_0,
author = {Patrick Ingram},
title = {A quantitative primitive divisor result for points on elliptic curves},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {609--634},
publisher = {Universit\'e Bordeaux 1},
volume = {21},
number = {3},
year = {2009},
doi = {10.5802/jtnb.691},
zbl = {1208.11073},
mrnumber = {2605536},
language = {en},
url = {jtnb.centre-mersenne.org/item/JTNB_2009__21_3_609_0/}
}
Patrick Ingram. A quantitative primitive divisor result for points on elliptic curves. Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 609-634. doi : 10.5802/jtnb.691. https://jtnb.centre-mersenne.org/item/JTNB_2009__21_3_609_0/

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

 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 1863855 | Zbl 0995.11010

 A. Bremner, J. H. Silverman and N. Tzanakis, Integral points in arithmetic progression on ${y}^{2}=x\left({x}^{2}-{n}^{2}\right)$. J. Number Theory, 80 (2000). | MR 1740510 | Zbl 1009.11035

 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 1953049 | Zbl 1021.11001

 R. D. Carmichael,On the numerical factors of the arithmetic forms ${\alpha }^{n}±{\beta }^{n}$. Annals of Math. 2nd series, 15 (1914), 30–48 and 49–70. | JFM 45.1259.10

 G. Cornelissen and K. Zahidi, Elliptic divisibility sequences and undecidable problems about rational points. J. Reine Angew. Math. 613 (2007). | MR 2377127 | Zbl 1178.11076

 S. David, Minorations de formes linéaires de logarithmes elliptiques. Mém. Soc. Math. France No. 62 (1995). | EuDML 94914 | Numdam | MR 1385175 | Zbl 0859.11048

 G. Everest, G. McLaren, and T. Ward, Primitive divisors of elliptic divisibility sequences. J. Number Theory 118 (2006). | MR 2220263 | Zbl 1093.11038

 P. Ingram, Elliptic divisibility sequences over certain curves. J. Number Theory 123 (2007). | MR 2301226 | Zbl 1170.11010

 P. Ingram, Multiples of integral points on elliptic curves. J. Number Theory, to appear (arXiv:0802.2651v1) | MR 2468477 | Zbl pre05485801

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

 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 1070716 | Zbl 0712.11001

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

 B. Poonen, Characterizing integers among rational numbers with a universal-existential formula. (arXiv:math/0703907) | MR 2530851 | Zbl pre05573657

 K. F. Roth, Rational approximations to algebraic numbers. Mathematika 2 (1955). | MR 72182 | Zbl 0066.29302

 A. Schinzel, Primitive divisors of the expression ${A}^{n}-{B}^{n}$ in algebraic number fields. J. Reine Angew. Math. 268/269 (1974). | MR 344221 | Zbl 0287.12014

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

 J. H. Silverman, The arithmetic of elliptic curves. Volume 106 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1986. | MR 817210 | Zbl 0585.14026

 J. H. Silverman, Wieferich’s criterion and the $abc$-conjecture. J. Number Theory 30 (1988). | MR 961918 | Zbl 0654.10019

 J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. Volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. | MR 1312368 | Zbl 0911.14015

 J. H. Silverman, Generalized greatest common divisors, divisibility sequences, and Vojta’s conjecture for blowups. Monatshefte für Mathematik 145 (2005). | MR 2162351 | Zbl pre02232178

 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 476628 | Zbl 0366.12002

 R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. Acta Arithmetica 67 (1994). | MR 1291875 | Zbl 0805.11026

 P. Vojta, Diophantine approximations and value distribution theory. Volume 1239 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1987 | MR 883451 | Zbl 0609.14011

 M. Ward, Memoir on elliptic divisibility sequences. Amer. J. Math. 70 (1948). | MR 23275 | Zbl 0035.03702

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