发表机构
College of Information and Network Engineering, Anhui Science and Technology University（安徽科技学院信息与网络工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Boyd关于导子11的Mahler测度猜想中的第三个恒等式(C3)，通过将分裂积分转化为正则积分并利用模单位公式精确计算，同时给出了族S_k的完整结构分析。
AI 中文摘要
Boyd在1998年关于两个变量多项式的Mahler测度与椭圆曲线L值之间猜想恒等式m(P)=r|L'(E,0)|的表格，从最小可能的导子N=11开始。第三个导子11恒等式(C3)涉及多项式S_0=y^2+(x^2+1)y+x^3，该多项式在单位环面上消失，并断言围绕分支切割的log|y|的有符号分裂积分I_split等于b_11=L'(E_11,0)。我们证明了(C3)。证明将分裂积分识别为沿Samart的有符号开链γ̃的正则积分，通过小分支弧β_0闭合为闭反不变环C'，并证明其同调类为2γ^-，其中γ^-生成H_1(E,Z)^-：周期比是先验整数，球算术(Arb)将其确定为2。通过Brunault证明的Siegel单位公式的直接正则计算——符号{x,y}是X_1(11)上的一对模单位，因此不需要Bloch菱形定理——然后得出I_split=b_11；符号在区间算术中得到认证，恒等式与数值一致到366位。对于族S_k=y^2+(x^2+kx+1)y+x^3，我们精确确定了环面交点和模单位/调和情形；进一步的Boyd型评估作为猜想记录。在k=-1（导子53）时，该机制可证明地失败；附录有条件地处理Samart的导子17类比。所有计算均可从附带的代码重现；每个认证步骤都在区间算术内执行。
英文摘要
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 $I_{\mathrm{split}}$ 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 $\tildeγ$, 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 $H_1(E,\mathbb{Z})^-$: 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 $I_{\mathrm{split}}=b_{11}$; 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.
Comments34 pages. Certification scripts available at https://github.com/huiminZheng-collab/boyd-conductor11 and archived at https://doi.org/10.5281/zenodo.21820650 . Research carried out with the assistance of the AI system Kimi (Moonshot AI); see the declaration in the article
