On elliptic curves whose conductor is a product of two prime powers
HTML articles powered by AMS MathViewer
- by Mohammad Sadek;
- Math. Comp. 83 (2014), 447-460
- DOI: https://doi.org/10.1090/S0025-5718-2013-02726-3
- Published electronically: June 14, 2013
- PDF | Request permission
Abstract:
We find all elliptic curves defined over $\mathbb {Q}$ that have a rational point of order $N,\;N\ge 4$, and whose conductor is of the form $p^aq^b$, where $p,q$ are two distinct primes and $a,b$ are two positive integers. In particular, we prove that Szpiro’s conjecture holds for these elliptic curves.References
- Yu. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75–122. With an appendix by M. Mignotte. MR 1863855, DOI 10.1515/crll.2001.080
- Yann Bugeaud, Maurice Mignotte, and Yves Roy, On the Diophantine equation $(x^n-1)/(x-1)=y^q$, Pacific J. Math. 193 (2000), no. 2, 257–268. MR 1755817, DOI 10.2140/pjm.2000.193.257
- Zhenfu Cao, Chuan I. Chu, and Wai Chee Shiu, The exponential Diophantine equation $AX^2+BY^2=\lambda k^Z$ and its applications, Taiwanese J. Math. 12 (2008), no. 5, 1015–1034. MR 2431876, DOI 10.11650/twjm/1500574244
- Henri Cohen, Number theory. Vol. I. Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, Springer, New York, 2007. MR 2312337
- J. Gebel, A. Pethö, and H. G. Zimmer, On Mordell’s equation, Compositio Math. 110 (1998), no. 3, 335–367. MR 1602064, DOI 10.1023/A:1000281602647
- Toshihiro Hadano, Remarks on the conductor of an elliptic curve, Proc. Japan Acad. 48 (1972), 166–167. MR 311668
- Toshihiro Hadano, On the conductor of an elliptic curve with a rational point of order $2$, Nagoya Math. J. 53 (1974), 199–210. MR 354673
- Dino Lorenzini, Torsion and Tamagawa numbers, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 5, 1995–2037 (2012) (English, with English and French summaries). MR 2961846, DOI 10.5802/aif.2664
- A. P. Ogg, Abelian curves of $2$-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143–148. MR 201436, DOI 10.1017/s0305004100039670
- A. P. Ogg, Abelian curves of small conductor, J. Reine Angew. Math. 226 (1967), 204–215. MR 210706, DOI 10.1515/crll.1967.226.204
- Alice Silverberg, Open questions in arithmetic algebraic geometry, Arithmetic algebraic geometry (Park City, UT, 1999) IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 83–142. MR 1860041, DOI 10.1007/bf03026854
- 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
- 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
Bibliographic Information
- Mohammad Sadek
- Affiliation: Department of Mathematics and Actuarial Science, American University in Cairo, New Cairo, Egypt 11835
- Email: mmsadek@aucegypt.edu
- Received by editor(s): February 27, 2012
- Published electronically: June 14, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 83 (2014), 447-460
- MSC (2010): Primary 14H52
- DOI: https://doi.org/10.1090/S0025-5718-2013-02726-3
- MathSciNet review: 3120599