I am currently trying to decipher Mazur's Eisenstein ideal paper (not a comment about his clarity, rather about my current abilities). One of the reasons I am doing that is that many people told me that the paper was somehow revolutionary and introduced a new method into number theory. 
Could you explain exactly what subsequent developments did the paper bring, what ideas in the paper were considered more-or-less original (at the time it was published), and exactly what difficulties did these ideas resolve that people failed to resolve before the paper was published (if any)?
To answer your question, Mazur's paper "Torsion Points and Complex multiplication" [1] introduced several important ideas into the study of arithmetic algebraic geometry and complex multiplication. The paper was published in 1972 and was groundbreaking for several reasons:

1. Computation of complex multiplication: Mazur introduced a method to compute complex multiplications explicitly using torsion points on elliptic curves. This allowed for the first
The Mazur's paper "Ideals in \(p\)-adic number fields and arithmetic of elliptic curves" (Annals of Mathematics, 1972) is indeed seminal in the development of \(p\)-adic number theory and its applications to arithmetic geometry. One can divide the impact of the paper into several areas:

1. New constructions and classification of \(p\)-adic Lie alge