Applications of the Chebotarev density theorem to elliptic curves

A natural question to ask in the study of ℓ-adic and mod-ℓ representations attached to elliptic curves over Q is what conditions guarantee two such representations will be isomorphic. Related to this question is if we suppose two such representations are not isomorphic, does there exist a 'certificate' which proves they are not isomorphic? One way to show that two representations are not isomorphic is to show that they have different trace values on a group element. For an elliptic curve E over Q and a prime p of good reduction, one defines the trace of Frobenius ap(E), which is an integer independent of ℓ and arises as a trace value. In this thesis, we are interested in giving upper bounds for the smallest prime p such that ap(E) ̸= ap(E′). Serre [30] gives a classical asymptotic bound, although the constants are quite large. Recent work of Mayle-Wang [18] provides an explicit bound with smaller constants. We expand on Mayle-Wang's work and further reduce the constants appearing in their result. Both methods use explicit forms of the Chebotarev density theorem which assume the generalized Riemann hypothesis.