I have been told that Wiles's proof of FLT was based on elliptic curves. But yesterday I read from Takeshi Saito's book "Fermat's Last Theorem Basic Tools" that there is so called generalized elliptic curve, kind of proper flat scheme and whose geometric fibers are either isomorphic to either elliptic curve over an algebraically closed field or a Néron $n$-gon for some positive integer $n$. Is is so that somewhere in Wiles's proof he used generalized elliptic curves rather that elliptic curves?

  • $\begingroup$ You want to know if Taniyama-Shimura conjecture holds for generalized elliptic curves? These generalized elliptic curves are just the cusps in the moduli space. $\endgroup$ – user40276 Dec 30 '14 at 23:20
  • $\begingroup$ I am no expert in this but the statement Wiles proved is that every semi-stable elliptic curve is modular. (Of course we now know it is true for all elliptic curves.) I don't know what generalised elliptic curve is but at the very least it wasn't part of the 'statement' he proved but may have been used in the proof. $\endgroup$ – Jack Yoon Dec 30 '14 at 23:21
  • $\begingroup$ @user40276 No that is not I wanted to know in this question althought it is a good question as well. I have understand that the Taniyama-Shimura conjecture was on the modularity of semistable elliptic curves over $\mathbb Q$. But did Wiles used generalized elliptic curves somewhere when he proved the semistable case of TS-conjecture? $\endgroup$ – mathenthusiastic Dec 30 '14 at 23:26
  • $\begingroup$ Certainly he may have used the compactification of the moduli stack of elliptic curves (where the cusps live and hence the generalized elliptic curves), but I don't know if he used explicitly some technique of generalized elliptic curves. I don't even no how would be representations associated to those elliptic curves that are cusps on moduli $\endgroup$ – user40276 Dec 30 '14 at 23:33

The answer is that, "yes", Wiles's argument involves generalized elliptic curves at some points, in so far as (a) it involves arguing on various modular curves $X_0(N)$ and $X_1(N)$, whose cusps have a moduli interpretation in terms of generalized elliptic curves (as noted by [user40276] in comments above), and (b) it involves studying the bad reduction properties of elliptic curves over $\mathbb Q$, which can be described by extending these curves to generalized elliptic curves over $\mathbb Z$.

But this is not something to focus on, unless you are already familiar with many or most other aspects of Wiles's argument; it is a comparatively minor point. Essentially every argument in the arithmetic geometry of elliptic curves from the 1970's on which uses modular curves as one of its tools, including for examples Mazur's paper on classifying torsion, the Gross--Zagier paper, and so on, uses the notion of generalized elliptic curves; they are just one of the standard notions that belong to this area of mathematics.

I don't remember now, but it may also be that the actual phrase "generalized elliptic curve" doesn't actually appear in Wiles's Annals paper; the notion may just be there implicitly, encoded in various properties of modular curves that he uses, or other papers that he cites. (The standard background paper for properties of modular curves, where the notion of generalized elliptic curves first appears, is the paper of Deligne and Rapport.)


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