Quantitative upper bounds related to an isogeny criterion for elliptic curves

Abstract For and elliptic curves defined over a number field , without complex multiplication, we consider the function counting nonzero prime ideals of the ring of integers of , of good reduction for and , of norm at most , and for which the Frobenius fields and are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that and are not potentially isogenous if and only if , we investigate the growth in of . We prove that if and are not potentially isogenous, then there exist positive constants , , and such that the following bounds hold: (i) ; (ii) under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) under GRH, Artin's Holomorphy Conjecture for the Artin ‐functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin ‐functions of number field extensions.