Large Galois images for Jacobian varieties of genus 3 curves

E-print/Working paper (2015)

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 ... [more ▼]

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). [less ▲]

Tame Galois realizations of $ GSp_4(\Bbb F_łl)$ over $\Bbb Q$

in International Mathematics Research Notices (2011), (9), 2028--2046

In this paper, we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic l> 3 as the Galois group of a tamely ramified Galois extension of Q. The strategy ... [more ▼]

In this paper, we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic l> 3 as the Galois group of a tamely ramified Galois extension of Q. The strategy is to consider the Galois representation ρ_l attached to the Tate module at l of a suitable abelian surface. We need to choose the abelian surfaces carefully in order to ensure that the image of ρ_l is large and simultaneously maintain a control on the ramification of the corresponding Galois extension. We obtain an explicit family of curves of genus 2 such that the Galois representation attached to the l-torsion points of their Jacobian varieties provides tame Galois realizations of the desired symplectic groups. [less ▲]