AI 中文总结
该研究计算函数域上二次特征族及 Kummer 框架下原始三次特征薄族的 murmuration 密度，关联其期望与超椭圆曲线模空间稳定同调，获二次 L 函数中心点非零结果。
AI 中文摘要
我们计算函数域上二次特征族的 murmuration 密度，证明该密度的期望与该族中 Frobenius 类高次幂迹产生的修正项一致，并确定了该修正项的大亏格渐近行为。该修正项可通过带一个标记 Weierstrass 点的亏格g超椭圆曲线模空间的稳定同调描述，我们还将其识别为单级密度中的低阶项，通过避开比率猜想的替代方法得到了函数域上二次 L 函数在中心点的非零结果。此外，我们还计算了 Kummer 框架下原始三次特征的薄族的 murmuration 密度。
英文摘要
We compute the murmuration density for the family of quadratic characters over function fields. We show that the expectation of this density coincides with a correction term arising in the traces of high powers of the Frobenius class in this family, for which we determine large-genus asymptotics. This correction term admits a description in terms of the stable homology of the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point. We further identify this correction as a lower-order term in the one-level density and we obtain a non-vanishing result at the central point for quadratic $L$-functions over function fields, via an alternative approach to using the ratios conjecture. We also compute the murmuration density for the thin family of primitive cubic characters in the Kummer setting.
Comments24 pages
