An Arakelov-Theoretic Approach to Na\"ive Heights on Hyperelliptic Jacobians

Abstract

We use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the N\'eron-Tate height of the corresponding point on the Jacobian. We give an algorithm to compute the set of points of bounded height with respect to this new height. This provides an `in principle' solution to the problem of determining the sets of points of bounded N\'eron-Tate heights on the Jacobian. We give a worked example of how to compute the bound over a global function field for several curves, of genera up to 11.