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Mathematics > Algebraic Geometry

arXiv:2208.00017 (math)
[Submitted on 29 Jul 2022 (v1), last revised 2 Mar 2023 (this version, v2)]

Title:Computing heights via limits of Hodge structures

Authors: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.
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
Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
MSC classes: 11G50, 14D06, 14D07, 14J20, 14J70, 14Q15, 14Q20
Cite as: arXiv:2208.00017 [math.AG]
  (or arXiv:2208.00017v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.2208.00017

Submission history

From: Emre Can Sertöz [view email]
[v1] Fri, 29 Jul 2022 18:00:14 UTC (24 KB)
[v2] Thu, 2 Mar 2023 12:14:01 UTC (24 KB)
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