The arithmetic and geometry of genus four curves

We construct a point in the Jacobian of a non-hyperelliptic genus four curve which is defined over a quadratic extension of the base field. We attempt to answer two questions: 1. Is this point torsion? 2. If not, does it generate the Mordell--Weil group of the Jacobian? We show that this point generates the Mordell--Weil group of the Jacobian of the universal genus four curve. We construct some families of genus four curves over the function field of $\bP^1$ over a finite field and prove that half of the Jacobians in this family are generated by this point via the other half are not. We then turn to the case where the base field is a number field or a function field. We compute the Neron--Tate height of this point in terms of the self-intersection of the relative dualizing sheaf of (the stable model of) the curve and some local invariants depending on the completion of the curve at the places where this curve has bad or smooth hyperelliptic reduction. In the case where the reduction satisfies some certain conditions, we compute these local invariants explicitly