## Two idle questions about modular curves

This post is an math-blogging experiment in writing down small questions that have occurred to me, and which I haven’t thought about seriously — thus it is highly possible they are poorly formed, or that the answers are obvious.

1. Let f be a cuspform on S_2(Gamma_0(N)) such that A_f has dimension greater than 1. Then the map X_0(N) -> A_f factors through X_0(N)/W, where W is some group of Atkin-Lehner involutions which act as +1 on A_f. Do we know an example of such an f where the map X_0(N)/W -> A_f is not a closed embedding? What if dim A_f is greater than 2? (In some sense, a map from a curve to a three-fold should be less likely to intersect itself “by chance” than a map from a curve to a surface.)
2. In the original proof of Fermat’s Last Theorem, Mazur’s theorem on rational isogenies of elliptic curves over Q was used in a critical way; when E is your Frey curve, you prove that E is modular, then derive a contradiction from the fact that E[p] is an _irreducible_ modular mod p Galois representation with very little ramification. Nowadays, can one write down a proof of Fermat that doesn’t pass through Mazur’s theorem?