On the exponent of the group of points of an elliptic curve over a finite field
Author:
Francesco Pappalardi
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
Published electronically:
December 6, 2010
MathSciNet review:
2784798
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 where either
is fixed or
and
is prime. Here, we let both
and
vary; our estimate is explicit and does not depend on the elliptic curve.
- 1. 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
- 2. 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, https://doi.org/10.1016/j.crma.2003.10.006
- 3. 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
- 4. 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, https://doi.org/10.1155/IMRN.2005.1391
- 5. Kaisa Matomäki, A note on primes of the form 𝑝=𝑎𝑞²+1, Acta Arith. 137 (2009), no. 2, 133–137. MR 2491532, https://doi.org/10.4064/aa137-2-2
- 6. 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
- 7. 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
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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
DOI:
https://doi.org/10.1090/S0002-9939-2010-10658-5
Keywords:
Elliptic curves,
finite fields
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
Article copyright:
© Copyright 2010
American Mathematical Society