I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following: 



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> 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 $p$.
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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. 



Reference for the `standard' Tate curve argument.
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](http://www.math.wisc.edu/~ellenber/HBAVJan00.pdf). 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.


