Square-free orders for CM elliptic curves modulo p

Let E be an elliptic curve defined over Q, of conductor N , and with complex multiplication. We prove unconditional and conditional asymptotic formulae for the number of ordinary primes p ! N , p ≤ x , for which the group of points of the reduction of E modulo p has square-free order. These results are related to the problem of finding an asymptotic formula for the number of primes p for which the group of points of E modulo p is cyclic, first studied by Serre (1977). They are also related to the stronger problem about primitive points on E modulo p, formulated by Lang and Trotter (Bull Am Math Soc 83:289–292, 1977), and the one about the primality of the order of E modulo p, formulated by Koblitz [Pacific J. Math. 131(1):157–165, 1988]. Mathematics Subject Classification (2000) Primary 11G05; Secondary 11N36 · 11R45