Cyclicity of Elliptic Curves Modulo p and Elliptic Curve Analogues of Linnik’s Problem
نویسندگان
چکیده
1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.-P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O ( (log N) ) if E is without complex multiplication, and pE = O ( (log N) ) if E is with complex multiplication, for any 0 < ε < 1.
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