2024/07/19 by Zoé Yvon, Yvon, Zoé
Computer Science · Mathematics · #11F80 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Primary 11G05 #Secondary 11R32
paper · pdf · doi:10.48550/arxiv.2407.14370
openalex publication_date 2024/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an elliptic curve defined over a number field, the absolute Galois group acts on the group of torsion points of the elliptic curve, giving rise to a Galois representation in GL2(ℤ). The obstructions to the surjectivity of this representation are either local (i.e. at a prime), or due to nonsurjectivity on the product of local Galois images. In this article, we study an extreme case: the coincidence i.e. the equality of n-division fields, generated by the n-torsion points, attached to different positive integers n. We give necessary conditions for coincidences, dealing separately with vertical coincidences, at a given prime, and horizontal coincidences, across multiple primes, in particular when the Galois group on the n-torsion contains the special linear group. We also give a non-trivial construction for coincidences not occurring over ℚ.