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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: [email protected] 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: [email protected] 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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