Galois representations and Galois groups over Q

Sara Arias-De-Reyna (1, 2) , Cécile Armana (3) , Valentijn Karemaker (4) , Marusia Rebolledo (5) , Lara Thomas (3) , Núria Vila (6)

1 FSTC - Faculté des Sciences, de la Technologie et de la Communication
2 University of Luxembourg [Luxembourg]
3 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
4 Mathematical Institute
5 LMBP - Laboratoire de Mathématiques Blaise Pascal
6 UB - Universitat de Barcelona

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

Domaines: Théorie des nombres [math.NT]

Sara Arias-De-Reyna, Cécile Armana, Valentijn Karemaker, Marusia Rebolledo, Lara Thomas, et al.. Galois representations and Galois groups over Q. Women in Numbers Europe; Research Directions in Number Theory, 2, Springer International Publishing, 2015, Association for Women in Mathematics Series, 978-3-319-17987-2. ⟨10.1007/978-3-319-17987-2_8⟩. ⟨hal-01037959⟩