An analogue of the Siegel-Walfisz theorem for the cyclicity of CM elliptic curves mod \(p\)
type: Item
instance of: scholarly article
title: An analogue of the Siegel-Walfisz theorem for the cyclicity of CM elliptic curves mod \(p\)
zbMATH Open document ID: 1205.11061
author: Amir Akbary
author: V. Kumar Murty
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2010-07-09T00:00:00Z
review text: Let \(E\) be a CM elliptic curve defined over \({\mathbb Q}\) of conductor \(N\) and with complex multiplication by the full ring of integers of an imaginary quadratic field. Let \(C(x,N,E)\) be the number of primes \(p\leq x\) sucht that \(p\nmid N\) and the reduction \(\bmod\, p\) of \(E\) is cyclic. Let \(A,B>0\) and \(N\leq (\log x)^A\). The auhors prove that \[ C(x,N,E)={\mathfrak c}_E\text{li}(x)+O_{A,B}\left(\frac{x}{(\log x)^B}\right), \] uniformly in \(N\), where \[ {\mathfrak c}_E=\sum_{k=1}^\infty\frac{\mu(k)}{[{\mathbb Q}(E[k]):{\mathbb Q}]}, \] and the implied constant depends only on \(A\) and \(B\). Here \(\mu\) is the Möbius function. This is an analogue of the classical Siegel-Walfisz theorem on primes in arithmetic progressions.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 5734647
zbMATH Keywords: CM elliptic curves
zbMATH Keywords: reduction mod \(p\)
zbMATH Keywords: Siegel-Walfisz theorem
MaRDI profile type: Publication
full work available at URL: s13226-010-0002-4
OpenAlex ID: W2067066163
cites work: Q5503472
cites work: Cyclicity of CM elliptic curves modulo 𝑝
cites work: Cyclicity of elliptic curves modulo \(p\) and elliptic curve analogues of Linnik's problem
cites work: A Generalization of the Siegel-Walfisz Theorem
cites work: Cyclicity and generation of points mod \(p\) on elliptic curves
cites work: Q3856819
cites work: The Large Sieve Inequality for Algebraic Number Fields Iii. Zero-Density Results
cites work: Q4135671
cites work: Q4723876
cites work: On Artin's conjecture
cites work: Galois properties of points of finite order of elliptic curves
cites work: Q3710637
cites work: Q4317891
DOI: 10.1007/S13226-010-0002-4
description: scientific article; zbMATH DE number 5734647