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(𝔽ℓ) as a Galois group over ℚ for all primes ℓ∈ [ 11, 500, 000 ].
CITATION STYLE
Arias-de-Reyna, S., Armana, C., Karemaker, V., Rebolledo, M., Thomas, L., & Vila, N. (2015). Galois Representations and Galois Groups Over ℚ. In Association for Women in Mathematics Series (Vol. 2, pp. 191–205). Springer. https://doi.org/10.1007/978-3-319-17987-2_8
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