2
$\begingroup$

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the inertia subgroup $I_{v'}$ of $G_{F/\mathbb{Q}_p}$ is a subgroup of the automorphism group of $\widetilde{E}$. Here $\widetilde{E}$ is the elliptic curve reduced mod the unique maximal ideal of the ring of integers of $F$.

I would like a reference for this result, and I couldn't find it in Silverman's "The arithmetic of elliptic curves" (unless I missed it...). I would greatly appreciate any comments or a reference for this. Thank you.

Improve this question
$\endgroup$
7
  • 1
    $\begingroup$ What do you mean by the "smallest Galois extension over $\mathbb Q_p$?" Taken literally, the smallest Galois extension of any field is the field itself, and it is true that $G_{\mathbb Q_p/\mathbb Q_p}$ is a subgroup of $\text{Aut}(\tilde E)$, since it is the trivial group. But presumably that is not what you mean. $\endgroup$
    – Joe Silverman
    Commented Jan 27, 2018 at 21:48
  • 3
    $\begingroup$ You assume $E$ has potentially good reduction, and $F$ is the smallest Galois extension over which good reduction is attained (existence by Neron-Ogg-Shafarevich). The $F$-isomorphisms $f_g:g^{\ast}(E_F) \simeq E_F$ for $g \in G_{F/\mathbf{Q}_p}$ yield similar on Neron models. Passing to special fibers over the residue field $k$, the effect of $g^{\ast}$ disappears for $g \in I$, giving $k$-automorphisms $[g]$ of the reduction with $[gh]=[g][h]$ for $g,h\in I$. To show $[g]={\rm{id}}\Rightarrow g=1$, use minimality of $F$ and Neron-Ogg-Shafarevich with care. Works in any dimension. QED $\endgroup$
    – nfdc23
    Commented Jan 27, 2018 at 23:29
  • $\begingroup$ @nfdc23 This is what I was looking for thank you! $\endgroup$
    – Johnny T.
    Commented Jan 28, 2018 at 11:41
  • 1
    $\begingroup$ Oops, the premise has an error but can salvage well: there may not exist "the" (i.e., one) $F$ minimal over $\mathbf{Q}_p$ because having good reduction only depends on the compositum $F^{\rm{un}}$ with $\mathbf{Q}_p^{\rm{un}}$, an operation interacting poorly with "intersection" among $F$'s. For example, if $p>2$ then all ramified quadratic extensions of $\mathbf{Q}_p$ have the same $F^{\rm{un}}$, so if $E$ is a ramified quadratic twist of a good-reduction elliptic curve then no unique such $F$ exists. But we can make one with $F^{\rm{un}}$ the minimal option, and for this we win. $\endgroup$
    – nfdc23
    Commented Jan 28, 2018 at 12:30
  • 2
    $\begingroup$ For $\ell \ne p$, $\rho:I_p \to {\rm{GL}}(V_{\ell}(E))$ has open kernel corresponding to a finite Galois extension $K$ of $\mathbf{Q}_p^{\rm{un}}$ over which good reduction occurs (Neron-Ogg-Shafarevich), and it is the minimal option: good reduction occurs over a finite extension $F/\mathbf{Q}_p$ if and only if $F^{\rm{un}}\supset K$ (so $K$ is independent of $\ell$). Moreover, $K$ is Galois over $\mathbf{Q}_p$ since $\rho$ is the restriction to a normal subgroup of a representation of the Galois group of $\mathbf{Q}_p$, so there do exist finite Galois $F/\mathbf{Q}_p$ with $F^{\rm{un}}=K$. $\endgroup$
    – nfdc23
    Commented Jan 28, 2018 at 13:23

0

Reset to default

You must log in to answer this question.