Isomorphisms of Galois Representations of CM Elliptic Curves

Abstract

In this thesis, we consider the problem of determining all CM elliptic curves over $\rat$ whose mod $N$ Galois representations are isomorphic to the mod $N$ Galois representation of a given CM elliptic curve over $\rat$. By utilizing the results from the theory of complex multiplication as well as building on the works of Kani and Kwon on isogenies of CM elliptic curves and Zywina's work on the images of Galois representation over $\rat$, we are able to make substantial progress towards fully answering this problem.