$x^n+y^n=z^3$ What is Darmon-Merel's objection to using Conrad, Diamond, and Taylor?

Question

In "Winding quotients and some variants of Fermat’s Last Theorem" Darmon and Merel 2007 on page 4

They discuss the proof that

Assume that every elliptic curve over $Q$ is modular. Then $x^n+y^n = z^3$ has no non-trivial primitive solution when $n >= 3$.

They state:

In this case, our proof of part 3 of the Main Theorem still requires the hypothesis that the elliptic curves involved in the study of the equation $x^n+y^n = z^3$ are modular. This requirement is not a consequence of the results of Wiles, not even of the strengthenings due to Conrad, Diamond, and Taylor, since the conductor of these elliptic curves is divisible by $27$. The problem of showing that a cube cannot be expressed as a sum of two relatively prime nth powers ($n >= 3$) gives a Diophantine incentive for proving the entire Shimura-Taniyama conjecture [...]

My question is how on earth does the conductor being divisible by $27$ conflict with the modularity theorem? And how do you calculate it being divisible by $27$ in this case?

Answer 1

One possible confusion: The Darmon-Merel paper is from 1997 (not 2007), after Wiles and Conrad-Diamond-Taylor, but before the proof of the full modularity theorem of Breuil-Conrad-Diamond-Taylor 2001.

The modularity theorems for elliptic curves (generally) go via a p-adic modularity lifting theorem for $p = 3$, because $\mathrm{GL}_2(\mathbf{F}_3)$ representations are related to tetrahedral and octahedral Artin representations. One key technical aspect of Wiles' arguments is that one really needs to control the image of the $p$-adic Galois representation locally at $p$. When the representation (say coming from an elliptic curve) has level prime to $p$, this is not so hard --- either the representation is ordinary, or it is "finite flat." The higher the power of $p$, the more complicated the integral $p$-adic Hodge theory becomes. One should think of the main technical innovation of Conrad-Diamond-Taylor as understanding the technical issues for representations which become finite flat over a tamely ramified extension, and Breuil-Conrad-Diamond-Taylor as pushing these methods further to some wildy ramified cases. (This is why the Breuil-Conrad-Diamond-Taylor paper is subtitled "wild $3$-adic exercises" or something like that.) With B-C-D-T, one can handle the case when $27$ divides the conductor. Note that, up to twist, this is basically the largest power of $3$ which can arise from an elliptic curve over $\mathbf{Q}$. It should be said that there are further issues related to $p$-adic deformation rings for "potentially Barsotti-Tate" representations which required significant new methods, introduced by Kisin. So B-C-D-T were "lucky" in some sense that the case $27\| N$ turned out OK by their methods. (to be precise, the corresponding local deformation rings were smooth.)

The short answer: there is no "conflict" with the modularity theorem, it's just that earlier results used $p$-adic methods for $p = 3$ and required that the conductor be not too divisible by $p$. After the proof of the full modularity theorem by Breuil-Conrad-Diamond-Taylor, there were no longer any issues with $E/\mathbf{Q}$ of conductor divisible by $27$.

The question about computing the conductor is really answered in the article itself (Proposition 1.1)

(There is at the time of this answer a comment by reuns which completely irrelevant to the actual question, btw).

Answer 2

The answer by "Infinity" gets this completely right. The exact references here are:

• [CDT] Conrad, Brian; Diamond, Fred; Taylor, Richard. Modularity of certain potentially Barsotti-Tate Galois representations. J. Amer. Math. Soc. 12 (1999), no. 2, 521–567
• [BCDT] Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard. On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.

The [CDT] paper proves modularity of all elliptic curves which are "not too nasty" locally at the prime 3, i.e. have conductor not divisible by 27. The [BCDT] paper finishes the job by handling the remaining cases.

The Darmon--Merel paper was actually published in 1997, before either of these; but it seems that the results of [CDT] must have already been announced at that time, whereas the results of [BCDT] did not exist or had not been announced publicly.

(A source of confusion is the fact that the version of Darmon--Merel on Henri Darmon's web page here has a totally misleading date on it. This is presumably because Darmon wanted to make the paper available publicly once the journal's copyright had expired, for which he would have re-compiled the original LaTeX source code of the article, and the LaTeX rendering engine automatically date-stamped the file with the date it was compiled, not the date it was originally written. You will notice that all the references in the list at the end of this PDF are from 1996 or earlier.)