`-adic Representations and the Čebotarev Density Theorem

The Čebotarev Density Theorem, generalizing Dirichlet’s theorem on primes in arithmetic progression, gives us a notion of the density of prime ideals in a number field. We motivate our exposition of the theorem by giving a brief introduction to `-adic representations, as can be found in [Hus04]. We give a brief introduction to class field theory, drawing from [Cox89], and present Deuring’s [Deu35] simple proof of the Čebotarev Density Theorem. We conclude by following the impact of the Čebotarev Density Theorem on `-adic representations, by looking at `-adic representations attached to elliptic curves, one of the more famous instances being in the proof of Fermat’s Last Theorem.