发表机构
Université Marie et Louis Pasteur, Institut FEMTO-ST, CNRS（玛丽和路易·巴斯德大学，FEMTO-ST研究所，法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究椭圆曲线$L$-函数零点对BSD前导系数及Chebyshev竞赛交叉的影响，提出拓扑模型并证明硬边定律，实验揭示高度20处的定量分辨率障碍。
AI 中文摘要
我们研究了椭圆曲线$L$-函数的非中心零点如何进入Birch--Swinnerton-Dyer (BSD)前导系数以及椭圆Chebyshev竞赛中的素数尺度交叉。在椭圆曲线Riemann假设(ECRH)下，完备的$L$-函数给出精确的Hadamard分解，零点统计量$J_E$和$V_E$满足正恒等式$J_E=V_E/8+Δ_E$。对于导子$50700$的十七个秩一同源类，第一个零点平均贡献$Δ_{E,20}$的$82.4%$，而独立的边界值揭示了高度$20$以上的离散$35/36$零点计数分裂。对于Haar $SO(2N+1)$，我们证明了一个固定维数的硬边定律$\rm{Pr}{J_N>y}\sim C_Ne^{-3y/2}$。有限零点截断还在相位环面上定义了一个Morse函数，其水平集和余面积密度为交叉统计提供了拓扑模型。受假Chebyshev素数启发，我们通过对数加权Frobenius竞赛的中点和半跳来定义椭圆交叉素数。一个四阶段导子$50700$的计算显示在系综轮廓层面的一致性，而省略零点噪声阻碍了高度$20$处单个收缩窗口事件的解析。因此，该实验识别出一个定量分辨率障碍，而非素数采样转移的失败。
英文摘要
We study how the noncentral zeros of an elliptic-curve $L$-function enter the Birch--Swinnerton-Dyer (BSD) leading coefficient and the prime-scale crossings of an elliptic Chebyshev race. Under the elliptic-curve Riemann hypothesis (ECRH), the completed $L$-function yields an exact Hadamard decomposition, and the zero statistics $J_E$ and $V_E$ satisfy the positive identity $J_E=V_E/8+Δ_E$. For seventeen rank-one isogeny classes of conductor $50700$, the first zero contributes on average $82.4%$ of $Δ_{E,20}$, while independent edge values reveal a discrete $35/36$ zero-count split beyond height $20$. For Haar $SO(2N+1)$ we prove a fixed-dimensional hard-edge law $\Pr{J_N>y}\sim C_Ne^{-3y/2}$. A finite zero truncation also defines a Morse function on a phase torus, whose level sets and coarea density provide a topological model for crossing statistics. Motivated by false Chebyshev primes, we define elliptic crossing primes through the midpoint and half-jump of a logarithmically weighted Frobenius race. A four-stage conductor-$50700$ computation shows agreement at the level of ensemble profiles, while omitted-zero noise prevents resolution of individual shrinking-window events at height $20$. The experiment therefore identifies a quantitative resolution barrier rather than a failure of the prime-sampled transfer.
Comments34 pages, 5 figures, 7 tables
