Simply branched covers of curves and wild conductor exponents

Abstract

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 $C\to\mathbb{P}^1$.

Disclosure

“ion, so π is simple. Further, because (4 − 1)2 < 12, π is the unique gonal map of C by [15, Theorem 2]. Acknowledgements & Declarations. I thank Vladimir Dokchitser for his super- vision and Andrew Obus for reading a draft of this work. OpenAI’s ChatGPT suggested approaches to some technical lemmata herein, car- ried out literature searches and was used for some parts of the typesetting and proofreading process. However, all mathematical claims are verified by the author and this work is”

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Classification

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Verified

2026-08-23T19:10:18.451725Z

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