Semistable reduction of plane quartics at $p=3$

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 $X^+_{ns}(27)$.