## Prime power terms in elliptic divisibility sequences

HTML articles powered by AMS MathViewer

- by Valéry Mahé PDF
- Math. Comp.
**83**(2014), 1951-1991 Request permission

## Abstract:

We study a problem on specializations of multiples of rational points on elliptic curves analogous to the Mersenne problem. We solve this problem when descent via isogeny is possible by giving explicit bounds on the indices of prime power terms in elliptic divisibility sequences associated to points in the image of a nontrivial isogeny. We also discuss the uniformity of these bounds assuming the Hall–Lang conjecture.## References

- Mohamed Ayad,
*Points $S$-entiers des courbes elliptiques*, Manuscripta Math.**76**(1992), no. 3-4, 305–324 (French). MR**1185022**, DOI 10.1007/BF02567763 - 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), no. 2, 187–208. MR**1740510**, DOI 10.1006/jnth.1999.2430 - J. Cheon and S. Hahn,
*Explicit valuations of division polynomials of an elliptic curve*, Manuscripta Math.**97**(1998), no. 3, 319–328. MR**1654780**, DOI 10.1007/s002290050104 - D. V. Chudnovsky and G. V. Chudnovsky,
*Sequences of numbers generated by addition in formal groups and new primality and factorization tests*, Adv. in Appl. Math.**7**(1986), no. 4, 385–434. MR**866702**, DOI 10.1016/0196-8858(86)90023-0 - Capi Corrales-Rodrigáñez and René Schoof,
*The support problem and its elliptic analogue*, J. Number Theory**64**(1997), no. 2, 276–290. MR**1453213**, DOI 10.1006/jnth.1997.2114 - Gary Cornell and Joseph H. Silverman (eds.),
*Arithmetic geometry*, Springer-Verlag, New York, 1986. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR**861969**, DOI 10.1007/978-1-4613-8655-1 - Sinnou David,
*Minorations de formes linéaires de logarithmes elliptiques*, Mém. Soc. Math. France (N.S.)**62**(1995), iv+143 (French, with English and French summaries). MR**1385175** - Manfred Einsiedler, Graham Everest, and Thomas Ward,
*Primes in elliptic divisibility sequences*, LMS J. Comput. Math.**4**(2001), 1–13. MR**1815962**, DOI 10.1112/S1461157000000772 - Kirsten Eisenträger and Graham Everest,
*Descent on elliptic curves and Hilbert’s tenth problem*, Proc. Amer. Math. Soc.**137**(2009), no. 6, 1951–1959. MR**2480276**, DOI 10.1090/S0002-9939-08-09740-2 - Graham Everest, Patrick Ingram, Valéry Mahé, and Shaun Stevens,
*The uniform primality conjecture for elliptic curves*, Acta Arith.**134**(2008), no. 2, 157–181. MR**2429645**, DOI 10.4064/aa134-2-7 - Graham Everest, Victor Miller, and Nelson Stephens,
*Primes generated by elliptic curves*, Proc. Amer. Math. Soc.**132**(2004), no. 4, 955–963. MR**2045409**, DOI 10.1090/S0002-9939-03-07311-8 - Graham Everest and Thomas Ward,
*Heights of polynomials and entropy in algebraic dynamics*, Universitext, Springer-Verlag London, Ltd., London, 1999. MR**1700272**, DOI 10.1007/978-1-4471-3898-3 - M. Hindry and J. H. Silverman,
*The canonical height and integral points on elliptic curves*, Invent. Math.**93**(1988), no. 2, 419–450. MR**948108**, DOI 10.1007/BF01394340 - Patrick Ingram,
*Elliptic divisibility sequences over certain curves*, J. Number Theory**123**(2007), no. 2, 473–486. MR**2301226**, DOI 10.1016/j.jnt.2006.08.007 - Patrick Ingram,
*Multiples of integral points on elliptic curves*, J. Number Theory**129**(2009), no. 1, 182–208. MR**2468477**, DOI 10.1016/j.jnt.2008.08.001 - Patrick Ingram and Joseph H. Silverman,
*Uniform estimates for primitive divisors in elliptic divisibility sequences*, Number theory, Analysis and Geometry, Springer, New York, (2012), 243–271. - Serge Lang,
*Elliptic curves: Diophantine analysis*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. MR**518817**, DOI 10.1007/978-3-662-07010-9 - B. Mazur and J. Tate,
*The $p$-adic sigma function*, Duke Math. J.**62**(1991), no. 3, 663–688. MR**1104813**, DOI 10.1215/S0012-7094-91-06229-0 - D. W. Masser and G. Wüstholz,
*Estimating isogenies on elliptic curves*, Invent. Math.**100**(1990), no. 1, 1–24. MR**1037140**, DOI 10.1007/BF01231178 - Federico Pellarin,
*Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques*, Acta Arith.**100**(2001), no. 3, 203–243 (French). MR**1865384**, DOI 10.4064/aa100-3-1 - Clayton Petsche,
*Small rational points on elliptic curves over number fields*, New York J. Math.**12**(2006), 257–268. MR**2259240** - Bjorn Poonen,
*Hilbert’s tenth problem and Mazur’s conjecture for large subrings of $\Bbb Q$*, J. Amer. Math. Soc.**16**(2003), no. 4, 981–990. MR**1992832**, DOI 10.1090/S0894-0347-03-00433-8 - Takakazu Satoh,
*Generalized division polynomials*, Math. Scand.**94**(2004), no. 2, 161–184. MR**2053737**, DOI 10.7146/math.scand.a-14436 - Joseph H. Silverman,
*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210**, DOI 10.1007/978-1-4757-1920-8 - J. H. Silverman. Computing heights on elliptic curves.
*Math. Comp.*, 51(183):339–358, 1988. - Joseph H. Silverman,
*The difference between the Weil height and the canonical height on elliptic curves*, Math. Comp.**55**(1990), no. 192, 723–743. MR**1035944**, DOI 10.1090/S0025-5718-1990-1035944-5 - Joseph H. Silverman,
*Advanced topics in the arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR**1312368**, DOI 10.1007/978-1-4612-0851-8 - Joseph H. Silverman and Katherine E. Stange,
*Terms in elliptic divisibility sequences divisible by their indices*, Acta Arith.**146**(2011), no. 4, 355–378. MR**2747036**, DOI 10.4064/aa146-4-4 - Marco Streng,
*Divisibility sequences for elliptic curves with complex multiplication*, Algebra Number Theory**2**(2008), no. 2, 183–208. MR**2377368**, DOI 10.2140/ant.2008.2.183 - R. J. Stroeker and N. Tzanakis,
*Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms*, Acta Arith.**67**(1994), no. 2, 177–196. MR**1291875**, DOI 10.4064/aa-67-2-177-196 - N. Tzanakis and B. M. M. de Weger,
*How to explicitly solve a Thue-Mahler equation*, Compositio Math.**84**(1992), no. 3, 223–288. MR**1189890** - Jacques Vélu,
*Isogénies entre courbes elliptiques*, C. R. Acad. Sci. Paris Sér. A-B**273**(1971), A238–A241 (French). MR**294345** - P. M. Voutier and M. Yabuta,
*Lang’s conjecture for the elliptic curve $y^{2}~=~x\left ( x^{2}+ax\right )$*, International Journal of Number Theory**9**(2013), 1141–1170. - Samuel S. Wagstaff Jr.,
*Divisors of Mersenne numbers*, Math. Comp.**40**(1983), no. 161, 385–397. MR**679454**, DOI 10.1090/S0025-5718-1983-0679454-X - Morgan Ward,
*Memoir on elliptic divisibility sequences*, Amer. J. Math.**70**(1948), 31–74. MR**23275**, DOI 10.2307/2371930

## Additional Information

**Valéry Mahé**- Affiliation: École Polytechnique Fédérale de Lausanne, SB-IMB-CSAG, Station 8, CH-1015 Lausanne, Switzerland.
- Email: valery.mahe@epfl.ch
- Received by editor(s): December 24, 2009
- Received by editor(s) in revised form: October 15, 2011, and October 31, 2012
- Published electronically: November 12, 2013
- Additional Notes: This work was supported by EPSRC grant EP/E012590/1, the Université de Montpellier 2, the Université de Franche-Comté and the École Polytechnique Fédérale de Lausanne. The author thanks Professor Everest, Professor Silverman, Professor Stevens and the anonymous referee for helpful discussions and comments.
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**83**(2014), 1951-1991 - MSC (2010): Primary 11G05, 11A41
- DOI: https://doi.org/10.1090/S0025-5718-2013-02790-1
- MathSciNet review: 3194137