Richelot-Brandt图的结构迹恒等式与认证谱
A structural trace identity and certified spectra for the Richelot-Brandt graph
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中文总结 AI 辅助
本文证明Richelot-Brandt图相关的$R(\pi)$迹的结构公式，提出主亏格重数猜想的符号精细化版本，经精确算术认证验证了11至149之间所有素数的该预测。
中文摘要 AI 辅助
设$B_2(2)$是判别式为$p$的二元四元数埃尔米特格主亏格上的2次Brandt算子。从几何上看，$B_2(2)$是超特殊主极化阿贝尔曲面上Richelot (2,2)-同源图的加权邻接算子，且与Atkin-Lehner对合$R(\pi)$可交换。我们证明了$R(\pi)$在主亏格上的迹的一个结构公式：对每个素数$p \geq 7$，该迹是一个显式提升贡献（由权2和4的椭圆新形式的Atkin-Lehner特征空间确定）与权3的仿模非提升空间的带符号亏格之和。证明逐项比较了Ibukiyama给出的主亏格与非主亏格迹求值的闭式，所得带符号亏格的闭式公式对每个素数给出了Fricke符号偏差$d(p) \geq 0$。接下来，我们提出Ibukiyama主亏格重数猜想的一个特征值符号精细化版本，它预测$\mathrm{charpoly}\\, B_2(2)$可分解为 Eisenstein、Saito-Kurokawa、异号Yoshida、V_a型及一般型块，并指定了每个块上的$R(\pi)$符号，特别预测每个V_a型对的两个成员被相反的$R(\pi)$特征值分离。最后，精确算术认证验证了$11 \leq p \leq 149$的所有素数的该预测，验证过程构造了Richelot矩阵、检查其加权图结构、将提升块与单独计算的椭圆数据匹配，并通过投影子迹认证了带符号同型重数。在$p=19$处，第一个经认证的V_a型对被$R(\pi)$分离；在$p=61$处，该图实现了一般型因子$x + 7$。
英文摘要
The degree-$2$ Brandt operator $B_2(2)$ on the principal genus of binary quaternion Hermitian lattices of discriminant $p$ is the weighted adjacency operator of the Richelot $(2,2)$-isogeny graph on superspecial principally polarized abelian surfaces, and commutes with an Atkin-Lehner involution $R(π)$. For every prime $p\ge7$ we prove that the trace of $R(π)$ is the sum of an explicit lift contribution from elliptic newforms of weights $2$ and $4$ and a signed defect of the weight-$3$ paramodular non-lift space; the closed formula for the defect yields the Fricke-sign bias $d(p)\ge0$ for every prime. We formulate an eigenvalue-sign refinement of Ibukiyama's principal-genus multiplicity conjectures: charpoly $B_2(2)$ factors into Eisenstein, Saito-Kurokawa, opposite-sign Yoshida, type-Va, and general-type blocks with specified $R(π)$-signs. The type-Va clause is a theorem for every prime: by the global lifting theorem of Roesner and Weissauer for inner forms anisotropic at the archimedean place, the weak packet of a general-type representation of $GU_2(B)$ is the full product of its local $L$-packets, each member occurring with multiplicity one, so both members of every type-Va pair occur and the type-Va block is an exact square split evenly by $R(π)$. For the Saito-Kurokawa and Yoshida blocks the refinement remains conjectural. Exact-arithmetic certificates, replayable from a frozen archive, verify the full prediction at every prime $11\le p\le149$: at $p=19$ the first type-Va pair is separated by $R(π)$, and at $p=61$ the graph realizes the general-type factor $x+7$.
