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An open‐image theorem for a general class of abelian varieties

Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Néron model of A/K has at least one fiber with potential toric dimension 1, then, for almost all rational primes ℓ, the Galois group of the splitting field of the ℓ‐torsion of A is GSp2g(ℤ/ℓ).