The 2-Primary Class Group of Certain Hyperelliptic Curves Original Research Article

Let q be an odd prime, e a non-square in the finite field Fq with q elements, image(T) an irreducible polynomial in Fq[T] and A the affine coordinate ring of the hyperelliptic curve y2=eimage(T) in the (y, T)-plane. We use class field theory to study the dependence on deg(image) of the divisibility by 2, 4, and 8 of the class number of the Dedekind ring A. Applications to Jacobians and type numbers of certain quaternion algebras are given.