Small discriminants of complex multiplication fields of elliptic curves over finite fields

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• 作者：Igor E. Shparlinski
• 关键词：elliptic curve ; complex multiplication field ; Frobenius discriminant ; 11G20 ; 11N32 ; 11R11
• 刊名：Czechoslovak Mathematical Journal
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：65
• 期：2
• 页码：381-388
• 全文大小：119 KB
• 参考文献：[1] A. C. Cojocaru: Questions about the reductions modulo primes of an elliptic curve. Number Theory (H. Kisilevsky et al., eds.). Papers from the 7th Conference of the Canadian Number Theory Association, University of Montreal, Canada, 2002, CRM Proc. Lecture Notes 36, American Mathematical Society, Providence, 2004, pp. 61鈥?9.
  [2] A. C. Cojocaru, W. Duke: Reductions of an elliptic curve and their Tate-Shafarevich groups. Math. Ann. 329 (2004), 513鈥?34.MATH MathSciNet View Article
  [3] A. C. Cojocaru, E. Fouvry, M. R. Murty: The square sieve and the Lang-Trotter conjecture. Can. J. Math. 57 (2005), 1155鈥?177.MATH MathSciNet View Article
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  [7] F. Luca, I. E. Shparlinski: Discriminants of complex multiplication fields of elliptic curves over finite fields. Can. Math. Bull. 50 (2007), 409鈥?17.MATH MathSciNet View Article
  [8] H. L. Montgomery: Topics in Multiplicative Number Theory. Lecture Notes in Mathematics 227, Springer, Berlin, 1971.MATH
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  [10] J. H. Silverman: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106, Springer, New York, 2009.MATH
  
• 作者单位：Igor E. Shparlinski (1)
  
  1. Department of Pure Mathematics, School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, 2052, Australia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Convex and Discrete Geometry
  Ordinary Differential Equations
  Mathematical Modeling and IndustrialMathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9141

文摘

We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field \(\mathbb{F}_p\) of p elements, such that the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over \(\mathbb{F}_p\) is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I.E. Shparlinski (2007).