PreprintarXiv licencearXiv
Average twin prime conjecture for elliptic curves in arithmetic progressions
Abstract
In 1988, Koblitz conjectured an asymptotic formula for the number of primes p ≤ x for which the reduction of an elliptic curve over ℚ has prime order. Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes p lying in arithmetic progressions. Our asymptotic formula holds uniformly for moduli up to a fixed power of log x. We also show that the resulting average constant matches the theoretical constant predicted by Lee, Mayle, and Wang in 2025 using Galois representations.
Related
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BibTeX
@misc{guloglu2026average,
title = {{Average twin prime conjecture for elliptic curves in arithmetic progressions}},
author = {Ahmet M. Güloğlu and Asimina S. Hamakiotes and Sung Min Lee and Tian Wang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29938},
eprint = {2608.29938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29938},
}
RIS
TY - UNPB
TI - Average twin prime conjecture for elliptic curves in arithmetic progressions
AU - Ahmet M. Güloğlu
AU - Asimina S. Hamakiotes
AU - Sung Min Lee
AU - Tian Wang
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29938
UR - https://arxiv.org/abs/2608.29938
AB - In 1988, Koblitz conjectured an asymptotic formula for the number of primes p ≤ x for which the reduction of an elliptic curve over ℚ has prime order. Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes p lying in arithmetic progressions. Our asymptotic formula holds uniformly for moduli up to a fixed power of log x. We also show that the resulting average constant matches the theoretical constant predicted by Lee, Mayle, and Wang in 2025 using Galois representations.
ER -