On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences
Abdulmuhsin Alfaraj1
1 Department of Mathematical Sciences University of Bath Claverton Down Bath, BA2 7AY UK.
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 1, pp. 247-258.

Nous prouvons qu’il n’existe qu’un nombre fini de puissances parfaites dans les suites de divisibilité elliptiques générées par un point non entier sur une courbe elliptique de la forme y 2 =x(x 2 +b), où b est un entier positif non nul. Nous y parvenons en utilisant la modularité des courbes elliptiques sur les corps quadratiques réels.

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the form y 2 =x(x 2 +b), where b is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1244
Classification : 11B83, 11D61, 11G05
Mots clés : Modular methods, Elliptic divisibility sequences, Perfect powers
Abdulmuhsin Alfaraj 1

1 Department of Mathematical Sciences University of Bath Claverton Down Bath, BA2 7AY UK.
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2023__35_1_247_0,
     author = {Abdulmuhsin Alfaraj},
     title = {On the {Finiteness} of {Perfect} {Powers} in {Elliptic} {Divisibility} {Sequences}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {247--258},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {35},
     number = {1},
     year = {2023},
     doi = {10.5802/jtnb.1244},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1244/}
}
TY  - JOUR
AU  - Abdulmuhsin Alfaraj
TI  - On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2023
SP  - 247
EP  - 258
VL  - 35
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1244/
DO  - 10.5802/jtnb.1244
LA  - en
ID  - JTNB_2023__35_1_247_0
ER  - 
%0 Journal Article
%A Abdulmuhsin Alfaraj
%T On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences
%J Journal de théorie des nombres de Bordeaux
%D 2023
%P 247-258
%V 35
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1244/
%R 10.5802/jtnb.1244
%G en
%F JTNB_2023__35_1_247_0
Abdulmuhsin Alfaraj. On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 1, pp. 247-258. doi : 10.5802/jtnb.1244. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1244/

[1] Don Blasius; Jonathan D. Rogawski Motives for Hilbert modular forms, Invent. Math., Volume 114 (1993) no. 1, pp. 55-87 | DOI | MR | Zbl

[2] Wieb Bosma; John Cannon; Catherine Playoust The Magma Algebra System I: The User Language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl

[3] Yann Bugeaud; Maurice Mignotte; Samir Siksek Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. Math., Volume 163 (2006) no. 3, pp. 969-1018 | DOI | MR | Zbl

[4] Henri Carayol Sur les représentations ℓ-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 3, pp. 409-468 | DOI | MR | Zbl

[5] Henri Carayol Sur les représentations galoisiennes modulo attachées aux formes modulaires, Duke Math. J., Volume 59 (1989) no. 3, pp. 785-801 | MR | Zbl

[6] Henri Darmon Serre’s conjectures, Seminar on Fermat’s last theorem (CMS Conference Proceedings), Volume 17, American Mathematical Society, 1995, pp. 135-153 | MR | Zbl

[7] Fred Diamond; Jerry Shurman A first course in modular forms, Graduate Texts in Mathematics, 228, Springer, 2005, xv+436 pages

[8] Graham Everest; Helen King Prime powers in elliptic divisibility sequences, Math. Comput., Volume 74 (2005) no. 252, pp. 2061-2071 | DOI | MR | Zbl

[9] Graham Everest; Jonathan Reynolds; Shaun Stevens On the denominators of rational points on elliptic curves, Bull. Lond. Math. Soc., Volume 39 (2007) no. 5, pp. 762-770 | DOI | MR | Zbl

[10] Nuno Freitas; Bao V. Le Hung; Samir Siksek Elliptic Curves over Real Quadratic Fields are Modular, Invent. Math., Volume 201 (2015) no. 1, pp. 159-206 | DOI | MR | Zbl

[11] Nuno Freitas; Samir Siksek The Asymptotic Fermat’s Last Theorem for Five-Sixths of Real Quadratic Fields, Compos. Math., Volume 151 (2015), pp. 1395-1415 | DOI | MR | Zbl

[12] Nuno Freitas; Samir Siksek Criteria for irreducibility of mod p representations of Frey curves, J. Théor. Nombres Bordeaux, Volume 27 (2015) no. 1, pp. 67-76 | DOI | Numdam | MR | Zbl

[13] Kazuhiro Fujiwara Level optimisation in the totally real case (2006) (https://arxiv.org/abs/math/0602586v1)

[14] Patrick Ingram; Joseph H. Silverman Uniform estimates for primitive divisors in elliptic divisibility sequences, Number theory, analysis and geometry, Springer, 2012, pp. 243-271 | DOI | Zbl

[15] Frazer Jarvis Correspondences on Shimura curves and Mazur’s principle at p, Pac. J. Math., Volume 213 (2004) no. 2, pp. 267-280 | DOI | MR | Zbl

[16] Ron Livné Communication networks and Hilbert modular forms, Applications of algebraic geometry to coding theory, physics and computation (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 36, Kluwer Academic Publishers, 2001, pp. 255-270 | DOI | MR | Zbl

[17] Ali Rajaei On the levels of mod ℓ Hilbert modular forms, J. Reine Angew. Math., Volume 537 (2001), pp. 33-65 | MR | Zbl

[18] Jonathan Reynolds Perfect powers in elliptic divisibility sequences, J. Number Theory, Volume 132 (2012) no. 5, pp. 998-1015 | DOI | MR | Zbl

[19] Joseph H. Silverman The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 1986, xii+400 pages | DOI

[20] Joseph H. Silverman Wieferich’s criterion and the abc-conjecture, J. Number Theory, Volume 30 (1988) no. 2, pp. 226-237

[21] Joseph H. Silverman Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer, 1994, xiii+525 pages

[22] Marco Streng Elliptic divisibility sequences with complex multiplication, 2006 (Master’s thesis, Universiteit Utrecht)

[23] Richard Taylor On Galois representations associated to Hilbert modular forms, Invent. Math., Volume 98 (1989) no. 2, pp. 265-280

[24] Morgan Ward Memoir on elliptic divisibility sequences, Am. J. Math., Volume 70 (1948), pp. 31-74

[25] Andrew J. Wiles On ordinary λ-adic representations associated to modular forms, Invent. Math., Volume 94 (1988) no. 3, pp. 529-573

Cité par Sources :