Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation rho_{A, l}: Gal_Q -> GSp(6, l) attached to the l-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp(6, l).
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