Arithmetic Properties Related to Isogeny Criteria for Elliptic Curves and Drinfeld Modules

Let E_1 and E_2 be elliptic curves defined over a number field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of norm at most x for which E_1 and E_2 have the same Frobenius trace or Frobenius field is bounded above asymptotically by a function of x. We prove a bound with a log saving unconditionally, prove a bound with a power saving dependent on the Reimann hypothesis and generalized Reimann hypothesis, and prove a bound with a stronger power saving dependent on the generalized Reimann hypothesis, Artin's holomorphy conjecture, and a pair correlation conjecture. Let Phi_1 and Phi_2 be Drinfeld modules of rank r at least two defined over a generic A-field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of degree x for which Phi_1 and Phi_2 have the same Frobenius trace or characteristic polynomial of the Frobenius is bounded above asymptotically by a function of x. We prove bounds with a power saving dependent upon a generalization of the open image theorem for Drinfeld modules of Pink and Rütsche.