Images of 2-adic representations associated to hyperelliptic Jacobians

TextLet k   be a subfield of CC which contains all 2-power roots of unity, and let K=k(α1,α2,…,α2g+1)K=k(α1,α2,…,α2g+1), where the αiαi'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=12g+1(x−αi). Our main result states that the image of Galois in Sp(T2(J))Sp(T2(J)) coincides with the principal congruence subgroup Γ(2)◁Sp(T2(J))Γ(2)◁Sp(T2(J)). As an application, we find generators for the algebraic extension K(J[4])/KK(J[4])/K generated by coordinates of the 4-torsion points of J.VideoFor a video summary of this paper, please visit http://youtu.be/VXEGYxA6N8w.