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On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions  Nathan Jones
; Sung Min Lee
;  Date: 
2 Jun 2022  Abstract:  Let $E$ be an elliptic curve defined over $mathbb{Q}$ and, for a prime $p$
of good reduction for $E$ let $ ilde{E}_p$ denote the reduction of $E$ modulo
$p$. Inspired by an elliptic curve analogue of Artin’s primitive root
conjecture posed by S. Lang and H. Trotter in 1977, JP. Serre adapted methods
of C. Hooley to prove a GRHconditional asymptotic formula for the number of
primes $p leq x$ for which the group $ ilde{E}_p(mathbb{F}_p)$ is cyclic.
More recently, Akbal and G"{u}lo$reve{ ext{g}}$lu considered the question
of cyclicity of $ ilde{E}_p(mathbb{F}_p)$ under the additional restriction
that $p$ lie in an arithmetic progression. In this note, we study the issue of
which arithmetic progressions $a mod n$ have the property that, for all but
finitely many primes $p equiv a mod n$, the group
$ ilde{E}_p(mathbb{F}_p)$ is not cyclic, answering a question of Akbal and
G"{u}lo$reve{ ext{g}}$lu on this issue.  Source:  arXiv, 2206.00872  Services:  Forum  Review  PDF  Favorites 


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