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arXiv · 2208.00017

Computing heights via limits of Hodge structures

Spencer Bloch·Robin de Jong·Emre Can Sertöz
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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.

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Spencer Bloch, Robin de Jong, Emre Can Sertöz. 2022-07-29. Computing heights via limits of Hodge structures. https://arxiv.org/abs/2208.00017

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