1
$\begingroup$

Suppose $E\to \mathbb{G}_m/k$ is an elliptic curve with $k$ field of characteristic $p>0$ and $E[m]$ it $m$-torsion group with $(m,p)=1$.

Consider the induced finite etale cover $E[\ell]\to \mathbb{G}_m$.

Question: How to see that a sufficient condition for it to be tamely ramified in $0, \infty$ (ie that the induced finite map of regular proper models $\overline{E[\ell]} \to \mathbb{P}^1$ becomes tamely ramified in those points) is that $GL_2(\mathbb{Z}/\ell\mathbb{Z})$ has order prime to $p$.

The question is motivated by this answer.

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ The action of $\pi_1(\mathbf G_m)$ on a geometric fibre $E_{\bar x}[\ell]$ for $\bar x \colon \operatorname{Spec} \overline{\kappa(x)} \to \mathbf G_m$ is by linear automorphisms, so lands in $\operatorname{GL}_2(\mathbf Z/\ell\mathbf Z)$ (instead of merely the symmetric group on $\ell^2$ elements). Tamely ramified means that the action of the inertia subgroups at $0$, $1$, and $\infty$ factors through a quotient whose order is prime to $p$, but the whole group has order prime to $p$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 3 at 22:57

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .