Twists of genus three curves and their Jacobians

An algebraic curve is a curve defined over by polynomial equations with coefficients in a given field. This thesis treats problems which arise from genus three curves over finite fields. An important tool for treating such curves is the Jacobian variety of the curve. One problem is how many points such a curve or twists of such a curve can have. A second problem is when a twist of the associated Jacobian variety is itself a Jacobian. We treat both of these questions in this thesis. We do not always restrict to the case of finite fields. In particular the second question has a moduli interpretation in terms of the moduli spaces of genus three curves and principally polarised Abelian varieties. Thus despite the origin of the problems treated they quickly lead to more general and abstract parts of modern algebraic geometry. This contents of this thesis should therefore be of interest to both mathematicians working in the geometry over finite fields and mathematicians who are interested in the arithmetic of low dimensional moduli spaces of curves and Abelian varieties