Cyclicity and exponent of elliptic curves modulo $p$ in arithmetic progressions
Abstract
In this article, we study the cyclicity problem of elliptic curves modulo primes in a given arithmetic progression. We extend the recent work of Akbal and G\"ulo\u{g}lu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for CM elliptic curves , which also presents a generalisation of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty, and Serre. In addition, we refine the conditional estimates of Akbal and G\"ulo\u{g}lu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg, and Wu.
Keywords
Cite
@article{arxiv.2307.05594,
title = {Cyclicity and exponent of elliptic curves modulo $p$ in arithmetic progressions},
author = {Peng-Jie Wong},
journal= {arXiv preprint arXiv:2307.05594},
year = {2024}
}
Comments
Accepted by Q. J. Math