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The Square Sieve and the Lang–Trotter Conjecture

Published online by Cambridge University Press:  20 November 2018

Alina Carmen Cojocaru ,
Etienne Fouvry  and
M. Ram Murty
Alina Carmen Cojocaru
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey, USA 08544, email: cojocaru@math.princeton.edu
Etienne Fouvry
Affiliation:
Laboratoire de mathématiques, Université Paris-Sud Bât. 425, 9140 Orsay Cedex, France, CNRS UMR 8628, email Etienne.Fouvry@math.u-psud.fr
M. Ram Murty
Affiliation:
Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, Ontario, K7L 3N6, email: murty@mast.queensu.ca
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Abstract

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Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. Let $K$ be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes $p\,\le \,x$ for which $\mathbb{Q}\left( {{\pi }_{p}} \right)\,=\,K$, where ${{\pi }_{p}}$ denotes the Frobenius endomorphism of $E$ at $p$. More precisely, under a generalized Riemann hypothesis we show that this number is ${{O}_{E}}\left( {{x}^{17/18}}\,\log x \right)$, and unconditionally we show that this number is ${{O}_{E,K}}\left( \frac{x{{\left( \log \,\log x \right)}^{13/12}}}{{{\left( \log x \right)}^{25/24}}} \right)$ We also prove that the number of imaginary quadratic fields $K$, with − disc $K\,\le \,x$ and of the form $K\,=\,\mathbb{Q}({{\pi }_{p}})$, is ${{\gg }_{E}}\,\log \,\log \,\log \,x$ for $x\,\ge \,{{x}_{0}}\left( E \right)$. These results represent progress towards a 1976 Lang–Trotter conjecture.

Keywords

Primary: 11G05secondary: 11N3611R45Elliptic curves modulo pLang–Trotter conjectureapplications of sieve methods
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 6 , 01 December 2005 , pp. 1155 - 1177
Copyright
Copyright © Canadian Mathematical Society 2005

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