On the \({\ell}\) -adic valuation of the cardinality of elliptic curves over f

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• 作者：Josep M. Miret ; Jordi Pujolàs ; Javier Valera
• 关键词：11G20 ; Elliptic curve ; Finite field ; Group order ; $${\ell}$$ ?adic valuation
• 刊名：Archiv der Mathematik
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：105
• 期：3
• 页码：261-269
• 全文大小：494 KB
• 参考文献：1.R. Balasubramanian and N. Koblitz, The improbability that an elliptic curve has subexponential discrete log problem under the Menezes—Okamoto—Vanstone algorithm, J. Cryptology, 11 (1998), 141-45.
  2.D. A. Cox, Primes of the form x 2?+?ny 2, Wiley-Interscience, 1989.
  3.D. Freeman and K. Lauter, Computing endomorphism rings of Jacobians of genus 2 curves over finite fields, Proceedings of the First SAGA Conference, World Sci. Publ., (2008), 29-6.
  4.Ionica S., Joux A.: Pairing the volcano. Mathematics of Computation 82, 581-03 (2013)MathSciNet CrossRef MATH
  5.H. W Lenstra,. Jr., Complex multiplication structure of elliptic curves, Journal of Number Theory, 56 (1996), 227-41
  6.Menezes A.J., Okamoto T., Vanstone S.A.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans, Inform. Theory 39, 1639-646 (1993)MathSciNet CrossRef MATH
  7.J. Miret et?al., Volcanoes of \({\ell}\) -isogenies of elliptic curves over finite fields: the case \({\ell = 3}\) , Proceedings of the Primeras Jornadas de Teoría de Números, Publicacions Matemàtiques, (2007), 165-80.
  8.Rück H.-G.: A note on elliptic curves over finite fields. Mathematics of Computation 49, 301-04 (1987)MathSciNet CrossRef MATH
  9.Schoof R.: Nonsingular plane cubic curves over finite fields. Journal of Combinatorial Theory, Series A 46, 183-11 (1987)MathSciNet CrossRef MATH
  10.J. H. Silverman, The Arithmetic of Elliptic Curves, Second Edition, Graduate Texts in Mathematics 106, Springer, 2009.
  11.Voloch J.F.: A note on elliptic curves over finite fields. Bulletin de la S.M.F., 116, 455-58 (1988)MathSciNet MATH
  12.L. C. Washington, Introduction to Cyclotomic Fields, Second Edition, Graduate Texts in Mathematics 83, Springer, 1997.
  13.W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. école Norm. Sup. (4), 2 (1969), 521-60.
  14.Wittmann C.: Group structure of elliptic curves over finite fields. Journal of Number Theory 88, 335-44 (2001)MathSciNet CrossRef MATH
  
• 作者单位：Josep M. Miret (1)
  Jordi Pujolàs (1)
  Javier Valera (1)
  
  1. Departament de Matemàtica, Universitat de Lleida, 25001, Lleida, Spain
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8938

文摘

Let E be an elliptic curve defined over a finite field \({\mathbb{F}_{q}}\) of odd characteristic. Let \({\ell \neq 2}\) be a prime number different from the characteristic and dividing \({\# E(\mathbb{F}_{q})}\). We describe how the \({\ell}\)-adic valuation of the number of points grows by taking finite extensions of the base field. We also investigate the group structure of the corresponding \({\ell}\)-Sylow subgroups. Mathematics Subject Classification 11G20