AI 中文总结
研究\(p\)进域上曲线覆盖，利用形变理论通过\(p\)进扰动得到简单分支覆盖，在特定条件下可令\(C'\)与\(C\)相同，还给出了曲线在\(p>d\)时狂野导体指数公式，依据覆盖\(C\to\mathbb{P}^1\)分歧数据得出。
AI 中文摘要
我们运用形变理论证明，在\(p\)进域上光滑、射影且几何连通的曲线覆盖\(\pi\colon C\to D\)可进行\(p\)进扰动，以得到附近的简单分支覆盖\(\pi'\colon C'\to D\)。若\(D = \mathbb{P}^1\)且\(\pi^*\mathcal{O}_{\mathbb{P}^1}(1)\)非常充裕，则可令\(C' = C\)。作为应用，我们给出了曲线在素数\(p>d\)时的狂野导体指数公式，该公式依据任意次数\(d\)的覆盖\(C\to\mathbb{P}^1\)的分歧数据得出。
英文摘要
We use deformation theory to show that a cover of smooth, projective, geometrically connected curves $π\colon C\to D$ over a $p$-adic field can be $p$-adically perturbed to obtain a nearby simply branched cover $π'\colon C'\to D$ and that, if $D=\mathbb{P}^1$ and $π^*\mathcal{O}_{\mathbb{P}^1}(1)$ is very ample, then we may take $C'=C$. As an application, we give a formula for the wild conductor exponent of a curve at a prime $p>d$ in terms of the ramification data of any degree $d$ cover $C\to\mathbb{P}^1$.
Comments14 pages
