New York Journal of Mathematics
Volume 20 (2014) 927-957


David Holmes

An Arakelov-theoretic approach to naïve heights on hyperelliptic Jacobians

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Published: October 27, 2014
Keywords: Néron-Tate height, Arakelov theory, height-difference bounds, hyperelliptic curves
Subject: Primary: 11G30; secondary: 11G50, 37P30

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 Néron-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 Néron-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.

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Mathematisch Instituut Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands