Frey’s curve was an important link between Fermat’s Last Theorem and proving the modularity theorem. If there were a counterexample to Fermat’s Last Theorem, X^p + Y^p = Z^p, where p is an odd prime, then the elliptic curve with equation

would not be modular. So if all elliptic curves are modular, then no such X,Y,Z can exist, which thus proves Fermat’s Last Theorem.