A normalization issue in a transmitted form of Bloch's diamond formula: anti-invariant subgroup versus regulator quotient
Abstract
Bloch's identity ∫γη(f,g)=DE((f)diamond(g)), relating the regulator of a K2-symbol on an elliptic curve E/ℚ to the elliptic dilogarithm of the diamond product of the divisors, is transmitted in the Mahler-measure literature with γ generating the anti-invariant subgroup H1(E,ℤ)- and factor $1$ (e.g. Lalin-Ramamonjisoa, Thm. 6). We observe that, when E(ℝ) has one connected component (Δ<0), this reading is incompatible with the published intersection-product normalization (Brunault, Proposition 19): the anti-invariant subgroup then has index $2$ in the regulator quotient H1(E,ℤ)/H1(E,ℤ)+. The correct statement -- factor $1$ for the quotient generator, hence factor ι(E)∈{1,2} for the subgroup generator, with ι(E)=1⟺Δ>0 -- follows from Brunault's formula by an elementary index computation. We document the phenomenon twice: on 11.a3 by proved values (a Siegel-unit anchor from [Zheng, arXiv:2609.26119]; the diamond side by Brunault and Bertin), and on 20.a4 by high-precision computation of both integrals against the q-series for DE (subgroup/quotient ratio $2$ to $13$ digits). We also show that Touafek's Theorem 2 (involving a Δ<0 curve) mixes two independent normalizations (subgroup versus quotient, and a (2π)−1 prefactor versus the bare diamond value), and we identify the two-part reinterpretation making its printed formulas mutually compatible. In the cases we have checked, no Mahler-measure or DE-identity is affected: the main theorem of Lalin-Ramamonjisoa (conductor $17$, Δ>0) and Bertin's proved exotic relations stand untouched, and Touafek's final DE-relation is not contradicted and is numerically supported, though its printed derivation retains the gap described in the text. A systematic audit is beyond the scope of this note.
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BibTeX
@misc{zheng2026normalization,
title = {{A normalization issue in a transmitted form of Bloch's diamond formula: anti-invariant subgroup versus regulator quotient}},
author = {Huimin Zheng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05420},
eprint = {2611.05420},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05420},
}
RIS
TY - UNPB
TI - A normalization issue in a transmitted form of Bloch's diamond formula: anti-invariant subgroup versus regulator quotient
AU - Huimin Zheng
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.05420
UR - https://arxiv.org/abs/2611.05420
AB - Bloch's identity ∫_γη(f,g)=D_E((f)diamond(g)), relating the regulator of a K_2-symbol on an elliptic curve E/ℚ to the elliptic dilogarithm of the diamond product of the divisors, is transmitted in the Mahler-measure literature with γ generating the anti-invariant subgroup H_1(E,ℤ)^- and factor $1$ (e.g. Lalin-Ramamonjisoa, Thm. 6). We observe that, when E(ℝ) has one connected component (Δ<0), this reading is incompatible with the published intersection-product normalization (Brunault, Proposition 19): the anti-invariant subgroup then has index $2$ in the regulator quotient H_1(E,ℤ)/H_1(E,ℤ)^+. The correct statement -- factor $1$ for the quotient generator, hence factor ι(E)∈{1,2} for the subgroup generator, with ι(E)=1⟺Δ>0 -- follows from Brunault's formula by an elementary index computation. We document the phenomenon twice: on 11.a3 by proved values (a Siegel-unit anchor from [Zheng, arXiv:2609.26119]; the diamond side by Brunault and Bertin), and on 20.a4 by high-precision computation of both integrals against the q-series for D_E (subgroup/quotient ratio $2$ to $13$ digits). We also show that Touafek's Theorem 2 (involving a Δ<0 curve) mixes two independent normalizations (subgroup versus quotient, and a (2π)^−1 prefactor versus the bare diamond value), and we identify the two-part reinterpretation making its printed formulas mutually compatible. In the cases we have checked, no Mahler-measure or D_E-identity is affected: the main theorem of Lalin-Ramamonjisoa (conductor $17$, Δ>0) and Bertin's proved exotic relations stand untouched, and Touafek's final D_E-relation is not contradicted and is numerically supported, though its printed derivation retains the gap described in the text. A systematic audit is beyond the scope of this note.
ER -