# Does Fermat's Last Theorem imply the modularity theorem?

The Wikipedia article on the proof of Fermat's Last Theorem has this sentence

If the link identified by Frey could be proven, then in turn, it would mean that a proof or disproof of either of Fermat's Last Theorem or the Taniyama–Shimura–Weil conjecture would simultaneously prove or disprove the other.

This suggests FLT and the modularity theorem are equivalent. While the fact that the modularity theorem implies FLT was a rather important part of Wiles' proof, I wasn't aware the reverse implication was true. Is it?

• The modularity theorem is true so anything implies it. What implies the modularity of L-functions of rational elliptic curves should be essentially that for every $j(\tau) \in \mathbb{Q}$, some $j(\frac{a\tau+b}{d})$ appears in the invariants of the jacobian of some modular curve. Nov 15, 2018 at 22:29

1) The Shimura-Taniyama-Weil conjecture (now a theorem of Wiles et. al.) states that any elliptic curve $$E$$ defined over $$\mathbf Q$$ is "modular". This means roughly that the Hasse-Weil L-function $$L_E(s)$$ attached to $$E$$ comes from an L-function L$$(f_E,s)$$ attached to a certain modular form $$f_E$$. More precisely, let $$f_E$$ be the inverse Mellin transform of $$(2\pi)^{-s}\Gamma(s)L_E(s)$$; then $$f_E\in S_2(\Gamma_0 (N))$$, where $$N$$ is the conductor of $$E$$ and $$f_E$$ is a Hecke form.
2) Suppose that the Fermat equation $$a^p+b^p=c^p$$ admits a non trivial solution and consider the elliptic curve $$E_{a,b,c}$$ defined over $$\mathbf Q$$ by the equation $$y^2=x(x-a^p)(x+b^p)$$ (Hellegouarch, 1969). Frey (1986) conjectured that $$E_{a,b,c}$$ could not be modular (see the appendix). Then (Mazur and) Ribet (1990) proved a deep theorem on modular representations of $$Gal(\bar {\mathbf Q}/\mathbf Q)$$ which implied Frey's conjecture.