导子-判别式不等式I：缓分歧循环覆盖
Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I
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中文总结 AI 辅助
本文证明了离散赋值域上Z/n-覆盖的P^1的导子-判别式不等式，推广了超椭圆曲线情形，并加强了Kohls关于Jacobian导子指数的结果。
中文摘要 AI 辅助
我们证明了在离散赋值域K上定义的所有Z/n-覆盖的P^1的导子-判别式不等式，其中K具有优良的赋值环O_K，且剩余域完美且特征不整除n，模去第三作者工作中的一些计算。具体而言，当这样的曲线X由y^n=f(x)给出，其中f(x)∈O_K[x]且n整除deg(f)，并且若X是其在O_K上的最小正则模型，则X的Artin导子的相反数被(n-1)v_K(disc(rad(f)))所上界。这是前两位作者关于超椭圆曲线的先前工作的直接推广，而该先前工作又推广了Ogg、Saito、Liu和第二作者的工作。当f为首一多项式时，这加强了Kohls的一个结果，即此类曲线的Jacobian的导子指数被(n-1)v_K(disc(rad(f)))所上界。
英文摘要
We prove conductor-discriminant inequalities for all $\mathbb{Z}/n$-covers of $\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$, modulo some calculations appearing in work of the third author (arXiv:2609.20585). Specifically, when such a curve $X$ is given by $y^n = f(x)$ with $f(x) \in\mathcal{O}_K[x]$ and $n\mid\text{deg}(f)$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$. This is a direct generalization of previous work of the first two authors on hyperelliptic curves, which in turn generalized work of Ogg, Saito, Liu, and the second author. When $f$ is monic, this strengthens a result of Kohls stating that the conductor exponent of the Jacobian of such a curve is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$.
发表机构
- Baruch College（巴鲁克学院）
- Boston University（波士顿大学）
- Graduate Center, City University of New York（纽约市立大学研究生中心）
机构由 AI 辅助整理，请以论文原文为准。
