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 somebody informed about these matters 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)?
Best Answer
First, Mazur's paper is arguably the first paper where the new ideas (and language) of the Grothendieck revolution in algebraic geometry were fully embraced and crucially used in pure number theory. Here are several notable examples: Mazur makes crucial use of the theory of finite flat group schemes to understand the behavior of the $p$-adic Tate modules of Jacobians at the prime $p$. He studies modular forms of level one over finite rings (which need not lift to characteristic zero when the residue characteristic is $2$ or $3$). He proves theorems about mod-$p$ modular forms using what are essentially comparison theorems between etale cohomology and de Rham cohomology, and many more examples. The proof of the main theorem ($\S5$, starting at page 156) is itself a very modern proof which fundamentally uses the viewpoint of $X_0(N)$ as a scheme.
Second, there are many beautiful ideas which have their original in this paper: it contains many of the first innovative ideas for studying $2$-dimensional (and beyond) Galois representations, including the link between geometric properties (multiplicity one) and arithmetic properties, geometric conceptions for studying congruences between Galois representations, understanding the importance of the finite-flat property of group schemes, and the identification of the Gorenstein property. There is a theoretical $p$-descent on the Eisenstein quotient when previously descents were almost all explicit $2$-descents with specific equations. It introduces the winding quotient, and so on.
Third, while it is a dense paper, it is dense in the best possible way: many of the small diversions could have made interesting papers on their own. Indeed, even close readers of the paper today can find connections between Mazur's asides and cutting edge mathematics. When Mazur raises a question in the text, it is almost invariably very interesting. One particular (great) habit that Mazur has is thinking about various isomorphisms and by pinning down various canonical choices identifies refined invariants. To take a random example, consider his exploration of the Shimura subgroup at the end of section 11. He finishes with a question which to a casual reader may as well be a throw-away remark. But this question was first solved by Merel, and more recently generalized in some very nice work of Emmanuel Lecouturier. Lecouturier's ideas then played an important role in the work of Michael Harris and Akshay Venkatesh. Again, one could give many more such examples of this. Very few papers have the richness of footnotes and asides that this paper does. Never forget that one of the hardest things in mathematics is coming up with interesting questions and observations, and this paper contains many great ones - it is bursting with the ideas of a truly creative mathematician.
Finally, the result itself is amazing, and (pretty much) remains the only method available for proving the main theorem (the second proof due to Mazur is very related to this one). To give a sense of how great the theorem is, note that if $E$ is a semistable elliptic curve, then either $E$ is isogenous to a curve with a $p$-torsion point, or $E[p]$ is absolutely irreducible. This result (added for clarity: explicitly, Mazur's Theorem that $E/\mathbf{Q}$ doesn't have a $p$-torsion point for $p > 7$) is crucially used in Wiles' proof of Fermat. One could certainly argue that without this paper (and how it transformed algebraic number theory) we would not have had Wiles' proof of Fermat, but it's even literally true that Mazur's theorem was (and remains so today, over 40 years later) an essential step in any proof of Fermat.