Cyclicity and exponents of CM elliptic curves modulo $p$ in short intervals
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- by Peng-Jie Wong PDF
- Trans. Amer. Math. Soc. 373 (2020), 8725-8749
Abstract:
Let $E$ be a CM elliptic curve defined over $\Bbb {Q}$. We establish an asymptotic formula for the number of primes $p$ for which the reduction modulo $p$ of $E$ is cyclic over short intervals. This extends previous work of Akbary, Cojocaru, M. R. Murty, V. K. Murty, and Serre. Also, in light of the work of Freiberg, Kim, Kurlberg, Liu, and Wu, we estimate the average exponent of $E$ and the second moment of the number of distinct prime divisors of exponents of $E$ in short intervals. The key new idea is the use of our short interval generalisation of the work of Huxley and Wilson on the Bombieri–Vinogradov theorem for number fields.References
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Additional Information
- Peng-Jie Wong
- Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada
- MR Author ID: 1211484
- Email: pengjie.wong@uleth.ca
- Received by editor(s): September 18, 2019
- Received by editor(s) in revised form: April 1, 2020
- Published electronically: September 29, 2020
- Additional Notes: The author is currently a PIMS Post-Doctoral Fellow at the University of Lethbridge.
- © Copyright 2020 Peng-Jie Wong
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8725-8749
- MSC (2010): Primary 11G05, 11N36, 11G15, 11R45, 11R44
- DOI: https://doi.org/10.1090/tran/8197
- MathSciNet review: 4177274