Torsion points on elliptic curves and tame semistable coverings

P. A. Helminck

Published 2018-02-28Version 1

In this paper, we study tame Galois coverings of semistable models that arise from torsion points on elliptic curves. These coverings induce Galois morphisms of intersection graphs and we express the decomposition groups of the edges in terms of the reduction type of the elliptic curve. To that end, we first define the reduction type of an elliptic curve $E/K(C)$ on a subgraph of the intersection graph $\Sigma(\mathcal{C})$ of a strongly semistable model $\mathcal{C}$. In particular, we define the notions of good and multiplicative reduction on subgraphs of the intersection graph of $\Sigma(\mathcal{C})$. After this, we show that if an elliptic curve has good reduction on an edge and $\mathrm{char}(k)\nmid{N}$, then the $N$-torsion extension is unramified above that edge, as in the codimension one case. Furthermore, we prove a combinatorial version of a theorem by Serre on transvections for elliptic curves with a non-integral $j$-invariant. Our version states that the Galois representation $\rho_{\ell}:G\rightarrow{\mathrm{SL}_{2}(\mathbb{F}_{\ell})}$ of an elliptic curve with multiplicative reduction at an edge $e$ contains a transvection $\sigma\in{I_{e'/e}}$ for prime numbers $\ell$ that do not divide the slope of the normalized Laplacian of the $j$-invariant.