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.

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    $\begingroup$ For the benefit of other people like me who got a headache trying to understand the one-sentence proof: The Tate curve is a p-adic analytic description of an elliptic curve as the group quotient Q_p^*/{q^n}, where q is some power of p times a number with unit norm. Since the curve has potentially multiplicative reduction, this power of p is nonzero. The l-torsion is then generated by l-th roots of unity together with an l-th root of q. The question is whether this l-th root of q has the same p-adic norm as an integer power of p. \Delta is a power series that looks like q. [out of space] $\endgroup$
    – S. Carnahan
    Oct 16 '09 at 4:09

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.

  • $\begingroup$ It turns out theorem 1.2 of the following paper proves the same result using Neron models instead of Tate curves: springerlink.com/content/01174u56lpw55562 $\endgroup$ Nov 17 '09 at 22:41
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    $\begingroup$ I guess the lesson is that if you're going to include this assertion in your paper you should number it 1.2. $\endgroup$
    – JSE
    Nov 18 '09 at 1:30
  • $\begingroup$ At least up to a transposition (and an extra 0 after the decimal): it is also 2.10 of Ribet & Stein's "Lectures on Serre's conjecture". $\endgroup$ Nov 19 '09 at 19:59

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