Cyclicity and exponents of CM elliptic curves modulo 𝑝 in short intervals

Wong, Peng-Jie

doi:10.1090/tran/8197

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.

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