An Explicit Theory of Heights for Hyperelliptic Jacobians of Genus Three

The goal of this paper is to take up the approaches used to deal with Jacobians and Kummer surfaces of curves of genus 2 by Cassels and Flynn [CF] and by the author [Sto1, Sto3] and extend them to hyperelliptic curves of genus 3. Such a curve C is given by an equation of the form y = f(x), where f is a squarefree polynomial of degree 7 or 8. We denote its Jacobian by J . Identifying points with their negatives on J , we obtain the Kummer variety of J . It is known that the morphism J → P given by the linear system |2Θ| on J (where Θ denotes the theta divisor) induces an isomorphism of the Kummer variety with the image of J in P; we denote the image by K ⊂ P. Our first task is to find a suitable basis of the Riemann-Roch space L(2Θ) and give explicit equations defining K, thereby completing earlier work by Stubbs [Stu], Duquesne [Duq] and Müller [Mü1, Mü3]. To this end, we make use of the canonical identification of J with X = Pic(C) and realize the complement of Θ in X as the quotient of an explicit 6-dimensional variety V in A by the action of a certain group Γ. This allows us to identify the ring of regular functions on X \Θ with the ring of Γ-invariants in the coordinate ring of V . In this way, we obtain a natural basis of L(2Θ), and we find the quadric and the 34 quartics that define K. The next task is to describe the maps K → K and SymK → SymK induced by multiplication by 2 and by {P,Q} 7→ {P +Q,P −Q} on J . We use the approach followed in [Sto1]: we consider the action of a double cover of the 2-torsion subgroup J [2] on the coordinate ring of P. This induces an action of J [2] itself on forms of even degree. We use the information obtained on the various eigenspaces and the invariant subspaces in particular to obtain an explicit description of the duplication map δ and of the add-and-subtract map on K. In analogy with [Sto1], we also obtain an estimate for the local ‘loss of precision’ under δ in terms of the valuation of the discriminant of f . This in turn leads to an explicit upper bound for the difference h−ĥ between the (logarithmic) naive and canonical heights on J over a number field. Such a bound is necessary if one wants to find generators of the full Mordell-Weil group when only a subgroup of finite index is known. We