Papers › On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions
On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions
Nathan Jones, Sung Min Lee
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
Let E be an elliptic curve defined over ℚ and, for a prime p of good reduction for E let Ẽₚ 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 ≤x for which the group Ẽₚ(𝔽ₚ) is cyclic. More recently, Akbal and G\"{u}loğlu considered the question of cyclicity of Ẽₚ(𝔽ₚ) under the additional restriction that p lie in an arithmetic progression. In this note, we study the issue of which arithmetic progressions a n have the property that, for all but finitely many primes p ≡a n, the group Ẽₚ(𝔽ₚ) is not cyclic, answering a question of Akbal and G\"{u}loğlu on this issue.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections