Cyclicity and exponent of elliptic curves modulo p in arithmetic progressions

2023/07/10 by Peng‐Jie Wong, Wong, Peng-Jie
Mathematics · #11G05 #11G15 #11N36 #11R45 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.05594

openalex publication_date 2023/07/10 · openalex created_date 2023/07/14 · openalex updated_date 2026/07/28

Abstract

In this article, we study the cyclicity problem of elliptic curves E/ℚ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for CM elliptic curves E, 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 Güloğlu, 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 E modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg, and Wu.

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