Images of 2-adic representations associated to hyperelliptic Jacobians

2014/10/10 by Yelton, Jeffrey
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1410.2668

Abstract

Let k be a subfield of ℂ which contains all 2-power roots of unity, and let K = k(α₁, α₂, ... , α_{2g + 1}), where the α_i's are independent and transcendental over k, and g is a positive integer. We investigate the image of the 2-adic Galois action associated to the Jacobian J of the hyperelliptic curve over K given by y² = ∏_{i = 1}^{2g + 1} (x - α_i). Our main result states that the image of Galois in Sp(T₂(J)) coincides with the principal congruence subgroup Γ(2) \lhd Sp(T₂(J)). As an application, we find generators for the algebraic extension K(J[4]) / K generated by coordinates of the 4-torsion points of J.

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