On the surjectivity of mod \ℓ representations associated to elliptic\n curves

2015/08/30 by David Zywina, Zywina, David · 1 citation
Mathematics · Computer Science · Arts and Humanities · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #Historical Studies and Socio-cultural Analysis

paper · pdf · doi:10.48550/arxiv.1508.07661

Abstract

Let E be an elliptic curve over the rationals that does not have complex\nmultiplication. For each prime \ℓ, the action of the absolute Galois group\non the \ℓ-torsion points of E can be given in terms of a Galois\nrepresentation \ρ_E,\ℓ colon\n\Gal(\\ℚ/\ℚ) \→ GL₂( mathbbF_\ℓ). An\nimportant theorem of Serre says that \ρ_E,\ℓ is surjective for all\nsufficiently large \ℓ. In this paper, we describe a simple algorithm based\non Serre's proof that can quickly determine the finite set of primes \ℓ for\nwhich \ρ_E,\ℓ is not surjective. We will also give some improved bounds\nfor Serre's theorem.\n

Citations

• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

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