Square-free discriminants of Frobenius rings

Let E be an elliptic curve over Q. It is well known that the ring of endomorphisms of Ep, the reduction of E modulo a prime p of ordinary reduction, is an order of the quadratic imaginary field Q(πp) generated by the Frobenius element πp. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of a2p − 4p, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function πE,r,h(x) counting the number of primes p up to x such that a2p − 4p is square-free and in the congruence class r modulo h. We give in this paper the precise asymptotic for πE,r,h(x) when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo p.