I will discuss some of Mazur's work about congruences between Eisenstein series and cusp forms, and then end with an application to class groups of fields $\mathbb{Q}(N^{1/p})$, where $N$ and $p$ are primes.  I will only assume some algebraic number theory.  In particular, nothing about modular forms will be assumed.
 

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions. In this talk, I will present work (joint with Rachel Newton) developing explicit methods to study this principle for non-Galois extensions. As a key application, I will describe how these methods can be used to characterize the HNP for extensions whose normal closure has Galois group A_n or S_n. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for multinorm equations as well as consequences in the statistics of these local-global principles.