Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers.
Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$
I want to know the results [or references] about the ramification of $\rho$ at different primes $p$.
For example, can we know $\rho$ is unramified/tamely remified/wildly ramified at $p$, given conditons such that $E$ has good/mlutipicative reduction at $\ell$ or $p$?
Best Answer
I might as well answer this.
The first, and most important result here is the Néron-Ogg-Shafarevich criterion (see eg Serre-Tate, Good reduction of Abelian varieties – or just Silverman for elliptic curves, maybe the Advanced Topics but I’m not sure):
Given an Abelian variety $A$ over a local field $K$ of residue characteristic $p$, the following are equivalent:
In particular, if $A$ is an elliptic curve with potentially good reduction, the image of inertia on $A[m]$ (for $m \geq 3$ coprime to $p$) doe not depend (up to group isomorphism) on the choice of $m$.
This is because in a $q$-adic ring $R$, $GL_2(R)$ does not contain a torsion element congruent to $1$ mod $q$ (or $4$ when $q=2$).
The direct consequences are the following:
if $p \neq \ell$ and $E$ has good reduction at $p$, then $E[\ell]$ is unramified at $p$.
if $E$ has potentially good reduction (after a field extension of ramification index $e \mid 24$) at $p$ and $\ell\nmid 2p$, then the image of inertia at $p$ in $\mathrm{Aut}(E[\ell])$ always has cardinality $e$.
In case 2), at least when $p \neq 2,3$, the value of $e \in \{1,2,3,4,6\}$ can be worked out from the minimal discriminant of $E$: a source is Serre’s Propriétés galoisiennes des points d’ordre fini. You may want to complete that anyway with Silverman’s Advanced Topics section on Tate’s algorithm.
Note that this description of the values of $e$ implies that $E[\ell]$ is, in this case, not wildly ramified at $p$ when $p>3$.
Even when $p=2,3$, this is still very standard, but I don’t know of a reference (maybe Alain Kraus’s PhD thesis).
What we haven’t discussed is the situation when $\ell = p$ (I’ll get to this one later) and the one when $j(E) \notin \mathbb{Z}_p$.
In this case, we can’t possibly hope to ever apply Néron-Ogg-Shafarevich, but we can apply the theory of the $p$-adic uniformization (see Silverman’s Advanced topics, again).
The upshot is that the action of $G_{\mathbb{Q}_p}$ on $E[\ell]$ is an explicit quadratic twist of its action on the subgroup of $\overline{\mathbb{Q}_p}^{\times}/q^{\mathbb{Z}}$ generated by a $\ell$-th root of $q$ and a $\ell$-th root of unity, where $q \in p\mathbb{Z}_p$ is determined by $j(E) = q^{-1}+744+196884q+\ldots$.
This twist is
In particular, assuming $\ell >2$, $E[\ell]$ is unramified at $p$ iff $E$ has multiplicative reduction at $p$ and $v_p(j(E))$ is divisible by $\ell$, and it is at worst tamely ramified when $p>2$.
Before I briefly discuss the case $\ell=p$, I want to make a final remark about the situations I haven’t fully described – possible wild ramification.
There is an arithmetic invariant of $E$ called the conductor and denoted as $N$. By Ogg’s formula (this is definitely Silverman again but I forgot where), it can related to the geometry of $E$ at primes of bad reduction. But its original meaning is Galois-theoretic.
Let $L/\mathbb{Q}_p$ be a finite Galois extension and let $G$ be its Galois group acting on a free $R$-module $V$ of finite rank, where $R$ is a principal ring where $p$ is invertible.
The Swan exponent of this representation is $Sw(V) := \sum_{i \geq 1}{\frac{\dim{V/V^{G_i}}}{[G_0:G_i]}}$. It is a (nontrivial!) result that $Sw(V)$ is an integer (the dimensions are well-defined because the $G_i$ for $i \geq 1$ are $p$-groups).
So it describes how higher ramification groups act on $V$, in a way.
The point is that for any primes $\ell >2$ (to be on the safe side) and $p\neq \ell$, if the Swan exponent of $E[\ell]$ at $p$ is nonzero, then $v_p(N)=2+Sw(E[\ell])$.
And we finally reach the much more difficult case $\ell=p$. For the sake of simplicity, I assume $p\geq 5$. But $p=2,3$ is discussed in the references, which are mostly Serre’s previously-mentioned paper, Serre’s 1985 Sur les représentations modulaires de degré $2$ de $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, and Alain Kraus’s PhD thesis.
Anyway, the determinant of $E[\ell]$ is the cyclotomic character mod $\ell$, so it’s always ramified at $\ell$. The question is how much.
If $E[\ell]$ has potentially good supersingular reduction, it turns out that the ramification is tame.
If $E[\ell]$ has potentially good ordinary reduction, this is not known in general, I think. Even for good ordinary reduction, Serre doesn’t seem to know much about this in the 1973 paper.
Finally, since $p >2$, to deal with potentially multiplicative reduction we are reduced to the p-adic uniformization, where the description given above works verbatim for $\ell=p$, but with a different outcome: the representation is then tamely ramified iff $v_p(j(E))$ is divisible by $p$.