Semistable reduction of plane quartics at $p=3$

TL;DR

This paper addresses the problem of computing semistable reduction for plane quartic curves over local fields with residue characteristic $p=3$ by exploiting a degree-$3$ covering $Y\to{\mathbb P}^1_K$ and the different function of Cohen–Temkin–Trushin to locate tame and interior regions. It derives an explicit formula for the different $\delta_\varphi$ on intervals and provides an algorithm to construct a potentially $\varphi$-semistable model, with the implementation in SageMath's MCLF package. The method is demonstrated on a quotient of the non-split Cartan modular curve $X^+_{ns}(27)$, yielding concrete stable reductions and illustrating the computational workflow. The results deliver a self-contained treatment for plane quartics in characteristic $3$, including explicit local-reduction formulas and a practical pipeline for obtaining semistable reductions, enabling arithmetic applications in this residue characteristic.

Abstract

We explain how to compute the semistable reduction of plane quartic curves over local fields of residue characteristic $p=3$. Our approach is based on finding suitable degree-$3$ coverings of the projective line by such plane quartics and on the different function of Cohen, Temkin, and Trushin associated to the analytifications of these coverings. In particular, we give an explicit formula for computing the different function on a given interval. The resulting algorithm for computing the semistable reduction of plane quartics is implemented in SageMath, and we illustrate it by determining the semistable reduction of a particular plane quartic at $p=3$ that arises as a quotient of the non-split Cartan modular curve $X^+_{ns}(27)$.

Figures (4)

• Figure 1: The tree in in $({\mathbb{P}}^1_K)^{\mathop{\mathrm{an}}\nolimits}$ spanned by the boundary points of the tame locus associated to the tame locus of the curve considered in Example \ref{['ex-worked-example-1']}. This figure is adapted from a figure in diss
• Figure 2: The tree in $({\mathbb{P}}_K^1)^{\mathop{\mathrm{an}}\nolimits}$ spanned by the boundary points of the tame locus associated to $Y$. This figure is from diss
• Figure 3: The stable reduction of the curve $Y$. This figure is from diss
• Figure 4: The tree in $({\mathbb{P}}_{\mathbb{K}}^1)^{\mathop{\mathrm{an}}\nolimits}$ spanned by the zeros of $\Delta_F$ and $\psi_2$. This figure is from diss

Theorems & Definitions (42)

• Definition 1.1
• Remark 1.2
• Proposition 1.3
• proof
• Definition 1.4
• Remark 1.5
• Definition 1.6
• Remark 1.7
• Definition 2.1
• Remark 2.2