# The Modularity Theorem and Serre's/Faltings's Isogeny Theorem

Earlier this year I completed my Masters dissertation on Andrew Wiles's proof of The Modularity Theorem for semistable elliptic curves as a precursor to the accepted proof of Fermat's Last Theorem. However, there was one step in the proof which has so far eluded my understanding.

Given a semistable elliptic curve $E$ over $\mathbb{Q}$ with $3$-adic representation $\rho_{E,3}$ we know that this representation is modular due to Wiles's work on deformations and the modularity of the mod-$3$ representation, $\bar{\rho}_{E,3}$, which in turn is due to the Langlands-Tunnell theorem.

Now, the version of modularity Wiles uses in his paper is, essentially,

$E$ is modular if there is a non-constant morphism $X_0(N) \rightarrow E$ for some natural number $N$.

In Wiles's proof (http://math.stanford.edu/~lekheng/flt/wiles.pdf) it states on page 542,

By Serre's isogeny theorem, $E$ is modular (in the sense of being a factor of the Jacobian of a modular curve).

This is the step which confuses me. The question I am asking is: can anyone explain in more detail why Serre's (/Faltings's) isogeny theorem tells us that since $\rho_{E,3}$ is modular, there is a non-constant morphism $X_{0}(N) \rightarrow E$?

• For reference: This question was originally asked on MSE. It was re-posted here with my recommendation. – Ali Caglayan Aug 14 '14 at 12:18
• Modularity of $\rho_{E,3}$ means that there is a Galois equivariant surjective morphism from the $3$-adic Tate module of a modular Jacobian $J_0(N)$ to that of $E$. The isogeny theorem then implies that $E$ is a factor of $J_0(N)$. Then compose the surjective homomorphism $J_0(N) \twoheadrightarrow E$ with an Abel-Jacobi embedding $X_0(N) \hookrightarrow J_0(N)$ (which generates). Is that what you are asking? (Conversely, non-constant morphisms from a curve $C$ factor through surjective homomorphisms $\mathrm{Jac}(C) \to E$ of abelian varieties.) – Vesselin Dimitrov Aug 14 '14 at 12:40
• Thank you. Your comment does answer my question so I appreciate that. Perhaps the reason that I had difficulty accepting this part of the proof originally was that I had focused on a different definition of a modular representation which is not immediately compatible with Jacobians; I could not see how my definition could fit in with the theorem. – Neil Addison Aug 14 '14 at 12:52
• Vesselin, you could post your comment as an answer. – Joël Aug 14 '14 at 14:09