On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions

Publication date: 2 June 2022

Abstract: Let

E

be an elliptic curve defined over

m a t h b b Q

and, for a prime

p

of good reduction for

E

let

i l d e 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, J-P. Serre adapted methods of C. Hooley to prove a GRH-conditional asymptotic formula for the number of primes

p l e q x

for which the group

i l d e E_{p} (m a t h b b F_{p})

is cyclic. More recently, Akbal and G"{u}lo

lu considered the question of cyclicity of

i l d e E_{p} (m a t h b b F_{p})

under the additional restriction that

p

lie in an arithmetic progression. In this note, we study the issue of which arithmetic progressions

have the property that, for all but finitely many primes

, the group

i l d e E_{p} (m a t h b b F_{p})

is not cyclic, answering a question of Akbal and G"{u}lo

lu on this issue.

Has companion code repository: https://github.com/ncjones-uic/acyclicreductions

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Galois representations (11F80)

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