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The weak Chinburg conjecture on Mahler measures
Abstract
For every negative fundamental discriminant −f and every k≥1, we construct a rational function Rf,2k∈ℚ(x1,…,x2k) and a constant rf,2k∈ℚ× such that m(Rf,2k)=rf,2kL'(χ−f,1−2k), where m denotes the logarithmic Mahler measure and χ−f is the quadratic Dirichlet character associated with −f. This proves the weak Chinburg conjecture. An independent construction using Bloch cycles yields a stronger result in two variables: there exists a nonzero polynomial Rf∈ℚ[x,y] satisfying m(Rf)=4wL'(χ−f,−1), where w is the number of roots of unity in ℚ(√(−f)).
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BibTeX
@misc{guo2026weak,
title = {{The weak Chinburg conjecture on Mahler measures}},
author = {Xuejun Guo and Zhengyu Tao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14479},
eprint = {2609.14479},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14479},
}
RIS
TY - UNPB
TI - The weak Chinburg conjecture on Mahler measures
AU - Xuejun Guo
AU - Zhengyu Tao
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.14479
UR - https://arxiv.org/abs/2609.14479
AB - For every negative fundamental discriminant −f and every k≥1, we construct a rational function R_f,2k∈ℚ(x_1,…,x_2k) and a constant r_f,2k∈ℚ^× such that m(R_f,2k)=r_f,2kL'(χ_−f,1−2k), where m denotes the logarithmic Mahler measure and χ_−f is the quadratic Dirichlet character associated with −f. This proves the weak Chinburg conjecture. An independent construction using Bloch cycles yields a stronger result in two variables: there exists a nonzero polynomial R_f∈ℚ[x,y] satisfying m(R_f)=4wL'(χ_−f,−1), where w is the number of roots of unity in ℚ(√(−f)).
ER -