Details
| Original language | English |
|---|---|
| Journal | Experimental mathematics |
| Publication status | E-pub ahead of print - 13 Apr 2023 |
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.
Keywords
- Beilinson–Bloch pairing, biextension, height, limit mixed Hodge structure, nodal singularity, Néron–Tate pairing, period
ASJC Scopus subject areas
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In: Experimental mathematics, 13.04.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Computing Heights via Limits of Hodge Structures
AU - Bloch, Spencer
AU - de Jong, Robin
AU - Sertöz, Emre Can
N1 - Funding Information: We thank Vasily Golyshev, Matt Kerr, Greg Pearlstein, Matthias Schütt, Duco van Straten, and the anonymous referees for helpful remarks. We thank Raymond van Bommel, David Holmes and Steffen Müller for sharing with us their code related to [] and for helpful discussions. We also acknowledge the use of Magma [] and SageMath [] for facilitating experimentation. The third author gratefully acknowledges support from MPIM Bonn.
PY - 2023/4/13
Y1 - 2023/4/13
N2 - 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.
AB - 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.
KW - Beilinson–Bloch pairing
KW - biextension
KW - height
KW - limit mixed Hodge structure
KW - nodal singularity
KW - Néron–Tate pairing
KW - period
UR - http://www.scopus.com/inward/record.url?scp=85152937671&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2208.00017
DO - 10.48550/arXiv.2208.00017
M3 - Article
AN - SCOPUS:85152937671
JO - Experimental mathematics
JF - Experimental mathematics
SN - 1058-6458
ER -