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December 2020 On Artin’s conjecture for CM elliptic curves
Cristian Virdol
Kyoto J. Math. 60(4): 1361-1371 (December 2020). DOI: 10.1215/21562261-2019-0064

Abstract

Consider E a CM elliptic curve over Q . Assume that rank Q E 1 , and let a E ( Q ) be a point of infinite order. For p a rational prime, we denote by F p the residue field at p . If E has good reduction at p , let E ¯ be the reduction of E at p , let a ¯ be the reduction of a ( modulo p ), and let a ¯ be the subgroup of E ¯ ( F p ) generated by a ¯ . Assume that Q ( E [ 2 ] ) = Q and Q ( E [ 2 ] , 2 1 a ) Q . Then in this article we obtain an asymptotic formula for the number of rational primes p , with p x , for which E ¯ ( F p ) / a ¯ is cyclic, and we prove that the number of primes p , for which E ¯ ( F p ) / a ¯ is cyclic, is infinite. This result is a generalization of the classical Artin’s primitive root conjecture, in the context of CM elliptic curves; that is, this result is an unconditional proof of Artin’s primitive root conjecture for CM elliptic curves. Artin’s conjecture states that, for any integer a ± 1 or a perfect square (or equivalently a ± 1 , and Q ( ± 1 , a ) = Q ( 1 [ 2 ] , 2 1 a ) Q ), there are infinitely many primes p for which a is a primitive root (mod p ), and an asymptotic formula for such primes is satisfied (this conjecture is not known for any specific a ).

Citation

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Cristian Virdol. "On Artin’s conjecture for CM elliptic curves." Kyoto J. Math. 60 (4) 1361 - 1371, December 2020. https://doi.org/10.1215/21562261-2019-0064

Information

Received: 15 July 2016; Revised: 31 October 2018; Accepted: 12 November 2018; Published: December 2020
First available in Project Euclid: 6 October 2020

MathSciNet: MR4175811
Digital Object Identifier: 10.1215/21562261-2019-0064

Subjects:
Primary: 11G10
Secondary: 11G15

Keywords: Artin’s conjecture , Elliptic curves

Rights: Copyright © 2020 Kyoto University

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Vol.60 • No. 4 • December 2020
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