发表机构
Universidad Nacional Autónoma de México（墨西哥国立自治大学）
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Sun关于Domb数的两个超同余猜想，通过模周期和复乘理论，在惰性素数处得到模p^2可除性，在分裂素数处得到模p^3同余。
AI 中文摘要
我们证明了Z.-W. Sun提出的关于Domb数的两个超同余命题。第一个涉及无权重截断和$$S_p=\sum_{n=0}^{p-1}D_n$$，其由$\mathbb{Q}(\sqrt{-15})$中判别式$-60$的阶的两个真理想类所控制。我们证明了Sun关于模$p^2$同余的猜想，并在分裂情形下获得了模$p^3$的更强公式。第二个涉及由$X^2-11X+1$关联的Lucas序列加权的Domb和。Sun的原始表述有例外反例$p=3$；排除该素数后，我们证明了修正后的陈述，包括当两个相关二次特征均为$+1$时的模$p^3$细化。两个结果均源于一个模机制。扭曲的Domb级数是$\Gamma_0(6)+\langle 3\rangle$上的模周期，并通过二次回拉与Zagier的零星$C$-序列相关联。在相关复乘的虚二次域中的惰性素数处，截断的$C$-周期是Hasse不变量，并在超奇异约化时消失；有限回拉则给出模$p^2$的可除性。在分裂素数处，显式构造的行列式为$1$或$3$的矩阵识别出Beukers单位根，从而得到模$p^3$的同余。这两个应用分别对应于判别式$-60$和$-156$的复乘阶。
英文摘要
We prove two supercongruence statements proposed by Z.-W. Sun for Domb numbers. The first concerns the unweighted truncated sum $$S_p=\sum_{n=0}^{p-1}D_n$$ and is governed by the two proper ideal classes of the order of discriminant $-60$ in $\mathbb{Q}(\sqrt{-15})$. We prove Sun's conjectured congruences modulo $p^2$ and obtain stronger formulas modulo $p^3$ in the split cases. The second concerns Domb sums weighted by the Lucas sequences associated with $X^2-11X+1$. Sun's original formulation has the exceptional counterexample $p=3$; after excluding this prime, we prove the corrected statement, including its modulo-$p^3$ refinement when both relevant quadratic characters are $+1$. Both results arise from one modular mechanism. The twisted Domb series is a modular period on $Γ_0(6)+\langle 3\rangle$ and is related by a quadratic pullback to Zagier's sporadic $C$-sequence. At primes inert in the imaginary quadratic field underlying the relevant complex multiplication, the truncated $C$-period is the Hasse invariant and vanishes at supersingular reduction; a finite pullback then gives divisibility modulo $p^2$. At split primes, explicitly constructed determinant-$1$ or determinant-$3$ matrices identify the Beukers unit root, yielding congruences modulo $p^3$. The two applications correspond respectively to orders of complex multiplication of discriminants $-60$ and $-156$.
