Do modular forms of higher weight occur in the proof of FLT?

Question

Fermat's last theorem is a consequence of a statement about weight two modular forms. Going through the long proof, does one ever encounter modular forms of higher weight?

Answer 1

No, one does not encounter modular forms of any weight other than $2$ in the proof of Fermat's Last Theorem. This is due to the interaction of the weight $2$ modularity condition and complex differentiation.

In fact, if $X_1(N)$ is the modular curve associated to the group $\Gamma_1(N)$. Then the cotangent space of $X_1(N)$ (over $\mathbb{C}$) can be identified with the space of cusp forms of weight $2$ on $\Gamma_1(N)$. It is in this way (viewing forms as cotangents) that modular forms become associated to the intrinsic geometric properties of the modular curve.