Quantitative upper bounds related to an isogeny criterion for elliptic curves

For E₁ and E₂ elliptic curves defined over a number field K, without complex multiplication, we consider the function FE₁, E₂(x) counting non-zero prime ideals p of the ring of integers of K, of good reduction for E₁ and E₂, of norm at most x, and for which the Frobenius fields Q(πₚ(E₁)) and Q(πₚ(E₂)) are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that E₁ and E₂ are not potentially isogenous if and only if FE₁, E₂(x) = o (x/log x), we investigate the growth in x of FE₁, E₂(x). We prove that if E₁ and E₂ are not potentially isogenous, then there exist positive constants κ(E₁, E₂, K), κ'(E₁, E₂, K), and κ''(E₁, E₂, K) such that the following bounds hold: (i) FE₁, E₂(x) < κ(E₁, E₂, K) x (loglog x)1/9 (log x)19/18; (ii) FE₁, E₂(x) < κ'(E₁, E₂, K) x6/7 (log x)5/7 under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) FE₁, E₂(x) < κ''(E₁, E₂, K) x2/3 (log x)1/3 under GRH, Artin's Holomorphy Conjecture for the Artin L-functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin L-functions of number field extensions.