发表机构
University of Chicago; Texas A&M University; Williams College; University of Michigan（芝加哥大学; 德克萨斯农工大学; 威廉姆斯学院; 密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Miller偏差猜想，针对具有非常数j不变量的单参数椭圆曲线族，通过分解二阶矩的上同调分量并利用Sato-Tate猜想及Chebotarev密度定理，证明p项系数平均为负，从而确立偏差猜想。
AI 中文摘要
我们研究了Miller关于定义在有理数域上且具有非常数j不变量的椭圆曲线的一般单参数族的偏差猜想。利用Michel发展的方法，我们将二阶矩视为不同上同调分量的和：一个主要的p^2项，一个来自射影直线上一阶上同调的p^{3/2}项，以及对应于奇异纤维的p阶或更低阶的项。通过假设广义Sato-Tate猜想，我们运用表示论和动机理论来证明p^{3/2}项的系数在极限下平均为零。然后，我们在不假设Sato-Tate的情况下独立分析低阶贡献。通过应用Chebotarev密度定理和分析Galois表示，我们证明了来自加性奇异纤维的p阶项的贡献也平均为零。因此，p项仅存的平均贡献是-B_pp，其中B_p表示判别式消失的“坏”点的数目。因为B_p平均到一个严格正值，所以p项的总系数在极限下平均到一个严格负值，从而为这些族建立了偏差猜想。
英文摘要
We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over $\mathbb{Q}$ with non-constant $j$-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main $p^2$ term, a $p^{3/2}$ term arising from the first cohomology over the projective line, and terms of order $p$ or lower corresponding to singular fibers. By assuming the generalized Sato-Tate conjecture, we employ representation and motive theory to demonstrate that the coefficient of the $p^{3/2}$ term averages to zero in the limit. We then independently analyze the lower-order contributions without assuming Sato-Tate. By applying the Chebotarev density theorem and analyzing Galois representations, we show that the contributions to the order $p$ term from additive singular fibers also average to zero. Consequently, the only remaining average contribution to the $p$ term is $-B_pp$, where $B_p$ denotes the number of ``bad'' points where the discriminant vanishes. Because $B_p$ averages to a strictly positive value, the overall coefficient of the $p$ term averages to a strictly negative value in the limit, thereby establishing the bias conjecture for these families.
