Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

Andrew Obus; Padmavathi Srinivasan

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arXiv · 1910.02589

Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

Andrew Obus·Padmavathi Srinivasan

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Abstract

We prove an inequality between the conductor and the discriminant 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 $y^2 = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $X$ is its minimal regular model over $\mathcal{O}_K$, 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.

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Andrew Obus, Padmavathi Srinivasan. 2019-10-07. Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic. https://arxiv.org/abs/1910.02589

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