2016/05/31 by Andrew V. Sutherland, David Zywina · 32 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Combinatorics #Conjugacy class #Discrete mathematics #Elliptic curve #Finite Group Theory Research #Galois group #Galois module #Mathematics #Modular curve #Prime (order theory) #Prime power #Pure mathematics #Schoof's algorithm #Torsion (gastropod) #math.NT #msc:11F80 #msc:11G05 #msc:14G35
paper · pdf · doi:10.2140/ant.2017.11.1199
published in Algebra & Number Theory 11(5), 1199-1229 (Mathematical Sciences Publishers) · two typos in the sup column of Table 1 have been corrected
openalex publication_date 2017/07/12 · arxiv created 2021/04/02 · arxiv updated 2021/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
For each open subgroup G of \GL2(\\ℤ) containing -I with full determinant, let XG/\ℚ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in G. Up to conjugacy, we determine a complete list of the 248 such groups G of prime power level for which XG(\ℚ) is infinite. For each G, we also construct explicit maps from each XG to the j-line. This list consists of 220 modular curves of genus 0 and 28 modular curves of genus 1. For each prime \ℓ, these results provide an explicit\nclassification of the possible images of \ℓ-adic Galois representations arising from elliptic curves over \ℚ that is complete except for a finite set of exceptional j-invariants. This is joint work with David Zywina.