2018/02/28 by Paul Alexander Helminck, P. A. Helminck, Helminck, P. A.
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Anatomy #Biology #Elliptic curve #FOS: Mathematics #Geometric and Algebraic Topology #Mathematics #Polynomial and algebraic computation #Pure mathematics #Supersingular elliptic curve #Torsion (gastropod) #math.AG
paper · pdf · doi:10.48550/arxiv.1803.00079
published in arXiv (Cornell University) (Cornell University)
arxiv created 2018/02/28 · openalex publication_date 2018/02/28 · arxiv updated 2018/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Σ(C) of a strongly semistable model C. In particular, we define the notions of good and multiplicative reduction on subgraphs of the intersection graph of Σ(C). After this, we show that if an elliptic curve has good reduction on an edge and char(k)\nmidN, 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 ρℓ:G→SL2(\mathbbFℓ) of an elliptic curve with multiplicative reduction at an edge e contains a transvection σ∈Ie'/e for prime numbers ℓ that do not divide the slope of the normalized Laplacian of the j-invariant.