Computing Heights via Limits of Hodge Structures

Experimental Mathematics, vol. 33, pp. 547–558

Abstract

We consider the problem of explicitly computing Beilinson–Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes, between the height of certain limit mixed Hodge structures and certain Beilinson–Bloch heights obtained from odd-dimensional hypersurfaces with a node. This congruence suggests a new method to compute Beilinson–Bloch heights. Here we explain how to compute the relevant limit mixed Hodge structures in practice, then apply our computational method to a nodal quartic curve and a nodal cubic threefold. In both cases we explain the nature of the primes occurring in the congruence.

Authors 3

1. Spencer Janney Bloch

   University of Chicago

   Affiliation as printed

   University of Chicago, Chicago, IL, USA

2. Robin de Jong Aachen

   Leiden University

   Affiliation as printed

   Universiteit Leiden, Leiden, The Netherlands

3. Emre Can Sertöz corresponding

   Leibniz University Hannover

   Affiliation as printed

   Institut für Algebraische Geometrie, Leibniz Universität Hannover, Hannover, Germany