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Let E be a CM elliptic curve defined over Q. We establish an asymptotic formula for the number of primes p for which the reduction modulo p of E is… Click to show full abstract
Let E be a CM elliptic curve defined over 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.
Journal Title: Transactions of the American Mathematical Society
Year Published: 2020
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