Boyd's conductor 30 Mahler measure conjectures

About 16 years old · traced to 2010

Let $F(b,c)$ denote the lattice sums used in the paper, and let $m(\alpha)$ and $g(\alpha)$ be the Mahler measures defined by

m(\alpha)=\int_0^1\int_0^1\log\left|y+y^{-1}+z+z^{-1}+\alpha\right|\,\mathrm d\theta_1\,\mathrm d\theta_2,

and

g(\alpha)=\int_0^1\int_0^1\log\left|(y+1)(z+1)(y+z)-\alpha yz\right|\,\mathrm d\theta_1\,\mathrm d\theta_2,

where $y=e^{2\pi i\theta_1}$ and $z=e^{2\pi i\theta_2}$ . Boyd's conductor 30 Mahler measure conjectures. The following formulas are numerically true:

\frac{15}{4\pi^2}F(2,15)=3m(1)-g(3),

\frac{15}{\pi^2}F\left(2,\frac{5}{3}\right)=3m(1)+g(3).

These formulas arise from an unproven conjectural relation between the conductor $30$ elliptic curve and $g(3)$ , together with a modular-equation relation among the lattice sums. The source provides numerical evidence but no proof or resolution.

References

Primary source

Mathew Rogers and Boonrod Yuttanan, “Modular equations and lattice sums”, arXiv:1001.4496 (2011).

Posted 1 month ago

Detchat Samart 1 month ago · edited 1 month ago
The conjectural identity between g(3) and the conductor 30 elliptic curve L-value has been proven (see Meemark, Y., Samart, D. Mahler measures of a family of non-tempered polynomials and Boyd’s conjectures. Res Math Sci 7, 1 (2020). https://doi.org/10.1007/s40687-019-0200-6.) Therefore, these conjectures are no longer open.

Progress summary

Refreshed 1 month ago

Claimed solved

A 2019 paper claims to prove both identities, but the claim has not been independently verified.

Boyd’s conductor-30 conjectures assert two exact identities linking four-dimensional lattice sums with Mahler measures. The 2010 source presented them as numerical conjectures associated with Rogers and Yuttanan.

Known results

In 2010, the conjectures had strong numerical support, but proofs were described as out of reach.
The original reduction used the conditional relation $L(E_{30},2)=\frac{2\pi^2}{15}g(3)$ together with a modular-equation identity for the lattice sums.

2019 claimed proof and September 6, 2026 community submission

Meemark and Samart’s 2019 paper states that its Mahler-measure identities prove all conductor-30 Boyd conjectures, including the two identities here. This is a claimed proof, not independently verified in the retrieved sources. A submission dated September 6, 2026 argues that later work by Meemark and Samart supplies the missing regulator argument and that normalization issues can be repaired, but this submission is also unverified.

Community submission (unverified)

The submission reconstructs the claimed proof using modular units, elliptic curves, $K$ -theory, diamond products, and elliptic regulators, emphasizing a change from a generalized Weierstrass model before applying the regulator identity.

Current status (as of September 2026): The two identities have a claimed 2019 proof, but no independent verification is recorded; the September 6, 2026 community submission remains unverified.

Sources

ProofManuscript status. This article is a rigorous reconstruction based on the uploaded 120-page audit document. The manuscript deliberately distinguishes identities verified algebraically, standard theorems invoked from the literature, and points requiring source-level reproducibility checks. Before journal submission, the bibliographic metadata and any computer-generated Fourier-coefficient tables should be matched against the original source PDFs.See full solutionHide full solution

We study the two conductor-30 identities conjectured by Rogers and Yuttanan for the four-dimensional lattice sum F(b,c). The identities relate F(2,15) and F(2,5/3) to the Mahler-measure quantity m(1) and the conductor-30 quantity g(3). The original argument reduced the problem to a second relation that was conjectural at the time. Later work of Meemark and Samart supplied a Mahler-measure proof using modular units, elliptic curves, K-theory, diamond products, and elliptic regulators. The present manuscript reconstructs that chain with special attention to the points at which normalization, torsion coordinates, Steinberg relations, and regulator signs can otherwise obscure the logic.

A central correction is made explicit: the torsion coordinates used in the published regulator calculation are naturally interpreted on a generalized Weierstrass model, and the change of model must be stated. Once this transformation is included, the divisor calculations and the resulting regulator identity are placed on a coherent elliptic-curve framework. We then combine the conductor-30 Mahler-measure relations with the Rogers–Yuttanan modular-equation identity and solve an invertible 2×2 linear system to obtain the two Boyd identities.

Conductor_30_Boyd_Corrected_Manuscript.pdf383,082 bytesOpen
boyds.pdf6,540,311 bytesOpen