Computing heights via limits of Hodge structures
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.
Cite
@article{arxiv.2208.00017,
title = {Computing heights via limits of Hodge structures},
author = {Spencer Bloch and Robin de Jong and Emre Can Sertöz},
journal= {arXiv preprint arXiv:2208.00017},
year = {2023}
}
Comments
17 pages. Meant as a follow-up to our previous paper (arXiv:2206.01220), we recall some results from there. New version incorporates referee comments. Final version