Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

Abstract

We prove a conductor-discriminant inequality for all hyperelliptic curves defined over discretely valued fields K with perfect residue field of characteristic not 2. Specifically, if such a curve is given by y, and if X is its minimal regular model over O K , then the negative of the Artin conductor of X (and thus also the number of irreducible components of the special fiber of X ) is bounded above by the valuation of disc(f ). There are no restrictions on genus of the curve or on the ramification of the splitting field of f . This generalizes earlier work of Ogg, Saito, Liu, and the second author.The proof relies on using so-called Mac Lane valuations to resolve singularities of arithmetic surfaces, a technique that was recently introduced by Wewers and the first author. Contents 1. Introduction Notation and conventions Acknowledgements 2. The discriminant bonus and regular models 3. A key reduction step 4. Mac Lane valuations 5. Mac Lane valuations, normal models and regular resolutions 6. Regular horizontal divisors 7. Discriminant bonus calculations 8. Comparing number of components and the discriminant bonus 9. Proof of the conductor-discriminant inequality in the irreducible case 10. Proof of the conductor-discriminant inequality in the reducible case Appendix A. Examples References

Integral differential forms for superelliptic curves

“…In this and the next section, there is quite some overlap with the articles [25] and [24], the reason being that our main assumptions and also our focus are somewhat different. For instance, in this article we can't afford to assume that the residue field k of our ground field K is algebraically closed, a very helpful assumption made in [25] and [24]. Another reason for being a bit repetetive is that we want to treat classical valuations and certain pseudovaluations on a equal footing.…”

“…We use the methods of [26] and [25]. See [24] for a slightly different treatment of essentially the same results.…”

A user's guide to the local arithmetic of hyperelliptic curves

“…We would like to mention some alternative techniques that have been recently developed for investigating similar topics. In [6,16,[21][22][23], the authors determine different kinds of models, the conductor exponent, the local 𝐿-factor, compare the Artin conductor to the minimal discriminant and compute a basis of the integral differentials. In arbitrary residue characteristic (including 2), but under some technical assumptions, [8,12,20] determine the minimal regular model with normal crossings, a basis of integral differentials, reduction types, conductor and action of the inertia group on the 𝓁-adic representation.…”

Regular models of hyperelliptic curves

“…In this section we summarise definitions and results on MacLane valuations. Our main references are [KW], [Mac], [OS1] and [Rüt].…”