On the exponent of the group of points of an elliptic curve over a finite field
HTML articles powered by AMS MathViewer
- by Francesco Pappalardi PDF
- Proc. Amer. Math. Soc. 139 (2011), 2337-2341 Request permission
Abstract:
We present a lower bound for the exponent of the group of rational points of an elliptic curve over a finite field. Earlier results considered finite fields $\mathbb {F}_{q^m}$ where either $q$ is fixed or $m=1$ and $q$ is prime. Here, we let both $q$ and $m$ vary; our estimate is explicit and does not depend on the elliptic curve.References
- Yann Bugeaud, Sur la distance entre deux puissances pures, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 12, 1119–1121 (French, with English and French summaries). MR 1396651
- William Duke, Almost all reductions modulo $p$ of an elliptic curve have a large exponent, C. R. Math. Acad. Sci. Paris 337 (2003), no. 11, 689–692 (English, with English and French summaries). MR 2030403, DOI 10.1016/j.crma.2003.10.006
- Florian Luca, James McKee, and Igor E. Shparlinski, Small exponent point groups on elliptic curves, J. Théor. Nombres Bordeaux 18 (2006), no. 2, 471–476 (English, with English and French summaries). MR 2289434
- Florian Luca and Igor E. Shparlinski, On the exponent of the group of points on elliptic curves in extension fields, Int. Math. Res. Not. 23 (2005), 1391–1409. MR 2152235, DOI 10.1155/IMRN.2005.1391
- Kaisa Matomäki, A note on primes of the form $p=aq^2+1$, Acta Arith. 137 (2009), no. 2, 133–137. MR 2491532, DOI 10.4064/aa137-2-2
- René Schoof, The exponents of the groups of points on the reductions of an elliptic curve, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 325–335. MR 1085266, DOI 10.1007/978-1-4612-0457-2_{1}5
- Lawrence C. Washington, Elliptic curves, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2008. Number theory and cryptography. MR 2404461, DOI 10.1201/9781420071474
Additional Information
- Francesco Pappalardi
- Affiliation: Dipartimento di Matematica, Università Roma Tre, Largo San Leonardo Murialdo 1, I–00146, Roma, Italy
- Email: pappa@mat.uniroma3.it
- Received by editor(s): December 23, 2009
- Received by editor(s) in revised form: June 13, 2010, and June 21, 2010
- Published electronically: December 6, 2010
- Communicated by: Ken Ono
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2337-2341
- MSC (2010): Primary 11G20; Secondary 11G05
- DOI: https://doi.org/10.1090/S0002-9939-2010-10658-5
- MathSciNet review: 2784798