Galois representations and Galois groups over Q

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve, let J(C) be the associated Jacobian variety and let ¯ρℓ : GQ → GSp(J(C)[ℓ]) be the Galois representation attached to the ℓ-torsion of J(C). Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes ℓ (if they exist) such that ¯ρℓ is surjective. In particular we realize GSp6 (Fℓ) as a Galois group over Q for all primes ℓ ∈ [11, 500000]