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 F₄䃑, ₄䃒 (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 (₏ (E₁) ) and Q (₏ (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 F₄䃑, ₄䃒 (x) = o (x x), we investigate the growth in x of F₄䃑, ₄䃒 (x). We prove that if E₁ and E₂ are not potentially isogenous, then there exist positive constants (E₁, E₂, K), ' (E₁, E₂, K), and '' (E₁, E₂, K) such that the following bounds hold: (i) F₄䃑, ₄䃒 (x) < (E₁, E₂, K) x (x) ^{1{9}} (x) ^{19{18}}; (ii) F₄䃑, ₄䃒 (x) < ' (E₁, E₂, K) x^{6{7}} (x) ^{5{7}} under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH) ; (iii) F₄䃑, ₄䃒 (x) < '' (E₁, E₂, K) x^2{3} (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.