Details
| Original language | English |
|---|---|
| Pages (from-to) | 547-558 |
| Number of pages | 12 |
| Journal | Experimental mathematics |
| Volume | 33 |
| Issue number | 4 |
| Early online date | 13 Apr 2023 |
| Publication status | Published - 2024 |
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
- Mathematics(all)
- General Mathematics
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In: Experimental mathematics, Vol. 33, No. 4, 2024, p. 547-558.
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 - Publisher Copyright: © 2023 Taylor & Francis Group, LLC.
PY - 2024
Y1 - 2024
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
VL - 33
SP - 547
EP - 558
JO - Experimental mathematics
JF - Experimental mathematics
SN - 1058-6458
IS - 4
ER -