Average Frobenius distributions for elliptic curves with 3-torsion
type: Item
instance of: scholarly article
title: Average Frobenius distributions for elliptic curves with 3-torsion
zbMATH Open document ID: 1158.11318
published in: Journal of Number Theory
publication date: 2005-02-22T00:00:00Z
review text: Let \(E/\mathbb Q\) be a non-CM elliptic curve and, for each prime \(p\), let \(a_p(E)=p+1-\#E(\mathbb F_p)\) be the trace of Frobenius. The Lang-Trotter conjecture says that \[ \pi_E^r(X):=\#\{p\leq X:\;a_p(E)=r\}\sim C(E,r)\sqrt{X}/\log X, \] where the constant \(C(E,r)\) is given explicitly in terms of the Galois representation attached to \(E\). \textit{E. Fouvry} and \textit{M. R. Murty} [Can. J. Math. 48, No. 1, 81--104 (1996; Zbl 0864.11030)] for \(r=0\) and \textit{C. David} and \textit{F. Pappalardi} [Int. Math. Res. Not. 1999, No. 4, 165--183 (1999; Zbl 0934.11033)] for general \(r\) proved that a version of the Lang-Trotter conjecture holds if one averages over all elliptic curves ordered by the size of their Weierstrass coefficients. Here the author examines the Lang-Trotter conjecture for elliptic curves which possess rational 3-torsion points. He proves that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 2136813
zbMATH Keywords: Elliptic curves
zbMATH Keywords: Lang-Trotter conjecture
zbMATH Keywords: Average Frobenius distributions
author: Kevin James
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.06.012
OpenAlex ID: W2038741633
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DOI: 10.1016/J.JNT.2004.06.012
description: scientific article; zbMATH DE number 2136813