Average lang-trotter conjecture for 2 elliptic curves

Let E 1 and E 2 be two elliptic curves without complex multiplication over the rationals. For primes p of good reduction, let a p ( E 1 ) and a p ( E 2 ) be the traces of the Frobenius morphism of E 1 and E 2 respectively. By Hasse's theorem, we know that a p ( E i ), i = 1,2, are integers and satisfy the inequalit