SQUARE-FREE DISCRIMINANTS OF FROBENIUS RINGS

Abstract

Let E be an elliptic curve over ℚ. We know that the ring of endomorphisms of its reduction modulo an ordinary prime p is an order of the quadratic imaginary field generated by the Frobenius element πp. However, except in the trivial case of complex multiplication, very little is known about the fields that appear as algebras of endomorphisms when p varies. In this paper, we study the endomorphism ring by looking at the arithmetic of [Formula: see text], the discriminant of the characteristic polynomial of πp. In particular, we give a precise asymptotic for the function counting the number of primes p up to x such that [Formula: see text] is square-free and in certain congruence class fixed a priori, when averaging over elliptic curves defined over the rationals. We discuss the relation of this result with the Lang–Trotter conjecture, and some other questions on the curve modulo p.