Tame Galois realizations of GSp₄(F_l) over Q

2009/10/08 by Sara Arias‐de‐Reyna, Sara Arias-de-Reyna, Arias-de-Reyna, Sara +2
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #math.NT #msc:11F80 #msc:11G30 #msc:11R32 #msc:12F12

paper · pdf · doi:10.48550/arxiv.0910.1445

29 pages

arxiv created 2009/10/08 · arxiv updated 2009/12/01

Abstract

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 varieties 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 provide tame Galois realizations of the desired symplectic groups.

Related

• Abelian varieties over number fields, tame ramification and big Galois image
• Galois representations and Galois groups over Q
• Constructing hyperelliptic curves with surjective Galois representations
• Tame torsion and the tame inverse Galois problem
• Large Galois images for Jacobian varieties of genus 3 curves