What is known that a proof of Fermat's Last Theorem 'would exist' at the time Andrew Wiles published?

Question

I heard in a conversation with a friend:

It was known prior to Wiles proof that a proof of Fermat's Last Theorem that a proof probably existed in the mathematics of eliptic curves.

Now when I read about the Modularity Theorem - I find a discussion about the Taniyama-Shimura Conjecture (potentially two names for the same thing).

Now it appears Frey suggested that Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem. This work eventually was called the Frey Curve.

But it sounds like other Mathematicians said at the time it was impossible. (Including Wiles ex-supervisor John Coates and Ken Ribet.)

My question is: What is known that a proof of Fermat's Last Theorem 'would exist' at the time Andrew Wiles published?

Answer 1

I'm no expert on the actual history or proof of Fermat's Last Theorem, but I'll try anyway.

According to a Numberphile video and Wikipedia, it was known that a consequence of proving the Taniyama-Shimura-Weil Conjecture was a proof of Fermat's Last Theorem. Simply, if you assumed Taniyama-Shimura was true, you had Fermat.

However, proving the conjecture seemed almost impossible. What Sir Andrew Wiles did was proving Taniyama-Shimura-Weil for a special case, which was later expanded to the more general case and became known as the Modularity Theorem, and then Fermat's became a corollary.