I work in number theory, a classical branch of mathematics which is primarily focused on understanding two things: the properties of prime numbers and solutions to polynomial equations. Using special geometric objects like elliptic curves and modular forms, for each prime number p we can construct p-adic Galois representations. Each of these Galois representations allows us to gain a little bit of information about all polynomial equations at once. I study various properties of these Galois representations. I’m especially interested in families of Galois representations which arise from geometric objects which are congruent mod p.