I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$, let $p$ be a prime, at which $E$ has potentially multiplicative reduction and let $\ell$ be a prime different than $p$. Then the mod $\ell$ representation is unramified at $p$ iff $\ell$ divides the valuation of $\Delta$ at $p$.
This is used for instance in the proof of Fermat's last theorem. In On modular representations of ${\rm Gal}(\overline{\mathbb Q}/\mathbb Q)$ arising from modular forms by Ken Ribet he cites Serre's (awesome) paper Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overline{\mathbb Q}/\mathbb Q)$, which (4.1.12) says this follows immediately from the theory of Tate curves.
It is pretty easy: the Tate curve gives you a explicit description the field obtained by adjoining the $\ell$-torsion points to $\mathbb Q_p$, and one can just check directly that the divisibility condition implies that this field (and thus the mod $\ell$ representation on the $\ell$-torsion points) is unramified at $p$.
Nonetheless I'm curious to know if anyone writes this down explicitly anywhere in the literature.
Best Answer
I think most people just mentally have in mind the argument you give.
In my thesis I actually wrote this down semi-carefully (including the case l = p, in which case what you want to say is that E[l] is finite over Zp, where E is now the Neron model of your elliptic curve over Q_p.) Or rather I wrote down the direction "l divides Delta => unramified" in Corollary 1.2 of the short version of my thesis. The goal of the thesis, by the way, was to extend this assertion to abelian varieties with real multiplication; the point being that it's not obvious what's supposed to play the role of Delta.