Boyd's conductor-11 Mahler measure conjecture: proof of the split-integral identity (C3), with an exact structural analysis of the family S_k
Abstract
Boyd's 1998 tables of conjectural identities m(P)=r |L'(E,0)| between Mahler measures of two-variable polynomials and L-values of elliptic curves begin with the smallest possible conductor, N=11. The third conductor-$11$ identity, (C3), concerns the polynomial S0=y2+(x2+1)y+x3, which vanishes on the unit torus, and asserts that a signed split integral Isplit of log|y| around the branch cut equals b11=L'(E11,0). We prove (C3). The proof identifies the split integral with a regulator integral along Samart's signed open chain γ̃, closed by a small-branch arc β0 into a closed anti-invariant cycle C', and proves its homology class is 2γ- with γ- generating H1(E,ℤ)-: the period ratio is a-priori integral, and ball arithmetic (Arb) pins it to $2$. A direct regulator computation via Brunault's proved Siegel-unit formula---the symbol {x,y} being a pair of modular units on X1(11), so Bloch's diamond theorem is not needed---then yields Isplit=b11; the sign is certified in interval arithmetic, and the identity agrees with the numerical value to $366$ digits. For the family Sk=y2+(x2+kx+1)y+x3 we determine exactly the torus intersections and the modular-unit/tempered cases; further Boyd-type evaluations are recorded as conjectures. At k=−1 (conductor $53$) the mechanism provably fails; an appendix treats Samart's conductor-$17$ analogue conditionally. All computations are reproducible from the accompanying code; every certification step is carried out within interval arithmetic.
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BibTeX
@misc{zheng2026boyds,
title = {{Boyd's conductor-11 Mahler measure conjecture: proof of the split-integral identity (C3), with an exact structural analysis of the family S_k}},
author = {Huimin Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26119},
eprint = {2609.26119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26119},
}
RIS
TY - UNPB
TI - Boyd's conductor-11 Mahler measure conjecture: proof of the split-integral identity (C3), with an exact structural analysis of the family S_k
AU - Huimin Zheng
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26119
UR - https://arxiv.org/abs/2609.26119
AB - Boyd's 1998 tables of conjectural identities m(P)=r |L'(E,0)| between Mahler measures of two-variable polynomials and L-values of elliptic curves begin with the smallest possible conductor, N=11. The third conductor-$11$ identity, (C3), concerns the polynomial S_0=y^2+(x^2+1)y+x^3, which vanishes on the unit torus, and asserts that a signed split integral I_split of log|y| around the branch cut equals b_11=L'(E_11,0). We prove (C3). The proof identifies the split integral with a regulator integral along Samart's signed open chain γ̃, closed by a small-branch arc β_0 into a closed anti-invariant cycle C', and proves its homology class is 2γ^- with γ^- generating H_1(E,ℤ)^-: the period ratio is a-priori integral, and ball arithmetic (Arb) pins it to $2$. A direct regulator computation via Brunault's proved Siegel-unit formula---the symbol {x,y} being a pair of modular units on X_1(11), so Bloch's diamond theorem is not needed---then yields I_split=b_11; the sign is certified in interval arithmetic, and the identity agrees with the numerical value to $366$ digits. For the family S_k=y^2+(x^2+kx+1)y+x^3 we determine exactly the torus intersections and the modular-unit/tempered cases; further Boyd-type evaluations are recorded as conjectures. At k=−1 (conductor $53$) the mechanism provably fails; an appendix treats Samart's conductor-$17$ analogue conditionally. All computations are reproducible from the accompanying code; every certification step is carried out within interval arithmetic.
ER -