Semistable abelian varieties with small division fields

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Let A be a semistable abelian variety defined over Q with bad reduction only at one prime p. Let L= Q(A[ℓ]) be the ℓ-division field of A for a prime ℓ not equal to p and let F= Q(μℓ) be the cyclotomic field generated by the group of ℓth-roots of unity. We study the varieties A for which H= Gal(L/F) is "small" in the sense that H is an ℓ-group or, more generally, that H is nilpotent. We show that if ℓ=2 or 3 and H is nilpotent then the reduction of A at p is totally toroidal, so its conductor is p A. The Jacobian of the modular curve X₀(41) is a simple semistable abelian variety of dimension 3, with bad reduction only at p=41 and the Galois group of its 2-division field is a 2-group. For ℓ=2, 3 or 5, there exist elliptic curves E of prime conductor such that Q(E[ℓ]) = Q(μ₂ ℓ). We characterize the abelian varieties that are isogenous to products Eᵈ.