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Average Frobenius distribution for the degree two primes of a number field

Published online by Cambridge University Press:  16 January 2013

KEVIN JAMES  and
ETHAN SMITH
KEVIN JAMES
Affiliation:
Department of Mathematical Sciences, Clemson University, Box 340975 Clemson, SC 29634-0975, U.S.A. e-mail: kevja@clemson.edu URL: www.math.clemson.edu/~kevja
ETHAN SMITH
Affiliation:
Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec, H3C 3J7, Canada; and Department of Mathematical Sciences, Michigan Technological University Townsend Drive, Houghton, Michigan, 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 number field and r an integer. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree two prime ideals of K with trace of Frobenius equal to r. Under certain restrictions on K, 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 154 , Issue 3 , May 2013 , pp. 499 - 525
Copyright
Copyright © Cambridge Philosophical Society 2013

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