On the surjectivity of mod ℓℓ representations associated to elliptic curves

2022/06/30 by David Zywina · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #Advanced Algebra and Geometry

paper · doi:10.1112/blms.12701

Abstract

Let E E be an elliptic curve over the rationals that does not have complex multiplication. For each prime ℓ ℓ , the action of the absolute Galois group on the ℓ ℓ -torsion points of E E can be given in terms of a Galois representation ρ E , ℓ : Gal ( Q ¯ / Q ) → GL 2 ( F ℓ ) ρ _E,ℓ \colon Gal(\mathbb Q/\mathbb Q) → GL₂(\mathbb F_ℓ ) . An important theorem of Serre says that ρ E , ℓ ρ _E,ℓ is surjective for all sufficiently large ℓ ℓ . In this paper, we describe a simple algorithm based on Serre's proof that can quickly determine the finite set of primes ℓ > 13 ℓ >13 for which ρ E , ℓ ρ _E,ℓ is not surjective. We will also give some improved bounds for Serre's theorem.

Citations

• Good Reduction of Abelian Varieties
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

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• Counting elliptic curves with prescribed entanglements

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