Elements in K_4 and regulator maps of Fermat curves
Abstract
We construct explicit elements in the group K_4^(3) of the Fermat curves x^N+y^N=1 for all N >= 3. The construction, which is uniform in N, uses polylogarithmic complexes and a map of de Jeu to K-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to 3/2 * ζ(3) * N^2 as N tends to infinity. Moreover, we derive formulas for the regulators of our elements in terms of hypergeometric functions, generalizing results of Otsubo for K_2 groups of Fermat curves. Finally, we numerically verify some cases of Beilinson's conjectures on special values of L-functions at s=3 for N = 3, 4, 6.
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BibTeX
@misc{brunault2026elements,
title = {{Elements in K_4 and regulator maps of Fermat curves}},
author = {François Brunault and David Lilienfeldt and Yusuke Nemoto},
howpublished = {HAL},
year = {2026},
month = jun,
eprint = {2606.24532},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05803673v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Elements in K_4 and regulator maps of Fermat curves
AU - François Brunault
AU - David Lilienfeldt
AU - Yusuke Nemoto
PY - 2026
DA - 2026/06/23
PB - HAL
UR - https://hal.science/hal-05803673v1
AB - We construct explicit elements in the group K_4^(3) of the Fermat curves x^N+y^N=1 for all N >= 3. The construction, which is uniform in N, uses polylogarithmic complexes and a map of de Jeu to K-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to 3/2 * ζ(3) * N^2 as N tends to infinity. Moreover, we derive formulas for the regulators of our elements in terms of hypergeometric functions, generalizing results of Otsubo for K_2 groups of Fermat curves. Finally, we numerically verify some cases of Beilinson's conjectures on special values of L-functions at s=3 for N = 3, 4, 6.
ER -