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 ρ̄ℓ: 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 ρ̄ℓ is surjective. In particular we realize GSp6(ℓ) as a Galois group over ℚ for all primes ℓ∈ [ 11, 500, 000 ]