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Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals

Published online by Cambridge University Press:  15 March 2011

KEVIN JAMES  and
ETHAN SMITH
KEVIN JAMES
Affiliation:
Department of Mathematical Sciences, Clemson University, Box 340975 Clemson, SC 29634-097, U.S.A. e-mail: kevja@clemson.edu URL: www.math.clemson.edu/~kevja
ETHAN SMITH
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931-1295, U.S.A. e-mail: ethans@mtu.edu URL: www.math.mtu.edu/~ethans
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Abstract

Let K be a fixed number field, assumed to be Galois over ℚ. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 3 , May 2011 , pp. 439 - 458
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Baier, S.The Lang-Trotter conjecture on average. J. Ramanujan Math. Soc. 22 (4): (2007), 299–314.Google Scholar
[2]Battista, J., Bayless, J., Ivanov, D. and James, K.Average Frobenius distributions for elliptic curves with nontrivial rational torsion. Acta Arith. 119 (1) (2005), 81–91.CrossRefGoogle Scholar
[3]Calkin, N., Faulkner, B., James, K., King, M. and Penniston, D. Average Frobenius distributions for elliptic curves over Abelian extensions. Acta Arith. (to appear).Google Scholar
[4]Cojocaru, A. C. and Ram Murty, M.An Introduction to Sieve Methods and Their Applications, London Mathematical Society Student Texts. vol. 66 (Cambridge University Press, 2006).Google Scholar
[5]David, C. and Pappalardi, F.Average Frobenius distributions of elliptic curves. Internat. Math. Res. Notices 1999 (4) (1999), 165–183.CrossRefGoogle Scholar
[6]David, C. and Pappalardi, F.Average Frobenius distribution for inerts in ℚ(i). J. Ramanujan Math. Soc. 19 (3) (2004), 181–201.Google Scholar
[7]Deuring, M.Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272.CrossRefGoogle Scholar
[8]Fouvry, E. and Ram Murty, M.On the distribution of supersingular primes. Canad. J. Math. 48 (1) (1996), 81–104.CrossRefGoogle Scholar
[9]Hecke, E.Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Math. Z. 1 (4) (1918), 357–376.CrossRefGoogle Scholar
[10]Hecke, E.Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Math. Z. 6 (1-2) (1920), 11–51.CrossRefGoogle Scholar
[11]Iwaniec, H. and Kowalski, E.Analytic Number Theory. American Mathematical Society Colloquium Publications. vol. 53 (American Mathematical Society, 2004).Google Scholar
[12]James, K.Average Frobenius distributions for elliptic curves with 3-torsion. J. Number Theory 109 (2) (2004), 278–298.CrossRefGoogle Scholar
[13]James, K.Averaging special values of Dirichlet L-series. Ramanujan J. 10 (1) (2005), 75–87.CrossRefGoogle Scholar
[14]James, K. and Yu, G.Average Frobenius distribution of elliptic curves. Acta Arith. 124 (1) (2006), 79–100.CrossRefGoogle Scholar
[15]Lang, S.Algebraic Number Theory. Graduate Texts in Mathematics. vol. 110 (Springer-Verlag, 1994).CrossRefGoogle Scholar
[16]Lang, S. and Trotter, H.Frobenius Distributions in GL2-extensions. Lecture Notes in Mathematics, Vol. 504. (Springer-Verlag, 1976). Distribution of Frobenius automorphisms in GL2-extensions of the rational numbers.CrossRefGoogle Scholar
[17]Lenstra, H.W. Jr.Factoring integers with elliptic curves. Ann. of Math. (2), 126 (3) (1987), 649–673.CrossRefGoogle Scholar
[18]Lidl, R. and Niederreiter, H.Finite Fields. Encyclopedia of Mathematics and its Applications. vol. 20 (Cambridge University Press, 1997). With a foreword by P. M. Cohn.Google Scholar
[19]Schoof, R.Nonsingular plane cubic curves over finite fields. J. Combin. Theory Ser. A 46 (2) (1987), 183–211.CrossRefGoogle Scholar
[20]Silverman, J. H.The Arithmetic of Elliptic Curves (Springer-Verlag, 1986).CrossRefGoogle Scholar
[21]Smith, E.A generalization of the Barban-Davenport-Halberstam Theorem to number fields. J. Number Theory 129 (11) (2009), 2735–2742.CrossRefGoogle Scholar
[22]Smith, E.A Barban-Davenport-Halberstam asymptotic for number fields. Proc. Amer. Math. Soc. 138 (7) (2010), 2301–2309.CrossRefGoogle Scholar
[23]Washington, L. C.Introduction to Cyclotomic Fields. Graduate Texts in Mathematics. vol. 83 (Springer-Verlag, 1997).CrossRefGoogle Scholar