Elliptic mod ℓ Galois representations which are not minimally elliptic

2004/09/07 by Dieulefait, Luis
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0409115

Abstract

In a recent preprint, F. Calegari has shown that for ℓ = 2, 3, 5 and 7 there exist 2-dimensional surjective representations ρ of \Gal(\Q/\Q) with values in \F_ℓ coming from the ℓ-torsion points of an elliptic curve defined over \Q, but not minimally, i.e., so that any elliptic curve giving rise to ρ has prime-to-ℓ conductor greater than the (prime-to-ℓ) conductor of ρ. In this brief note, we will show that the same is true for any prime ℓ >7, concretely, we will show that for any such ℓ the elliptic curve E^ℓ: Y² = X (X- 3^ℓ ) (X - 3^ℓ - 1) is semistable, has bad reduction at 3, the associated \mod ℓ Galois representation ρ is surjective, unramified at 3, and there is no elliptic curve with good reduction at 3 whose associated \mod ℓ representation is isomorphic to ρ.

Related

• On the surjectivity of mod \ℓ representations associated to elliptic\n curves
• On the surjectivity of mod ℓℓ representations associated to elliptic curves
• On the possible images of the mod ell representations associated to\n elliptic curves over Q
• Existence of non-elliptic mod l Galois representations for every l >5
• Mod p representations on elliptic curves