Wild conductor exponents of curves
Harry Spencer
TL;DR
The paper provides an explicit formula for the wild conductor exponent $n_{C,\text{wild}}$ of plane curves over a $p$-adic field $K$ in terms of invariants from explicit extensions of $K$, and shows that for genus $g$ with $p>g+1$ one can realize the same exponent on a hyperelliptic curve. Central to the approach is recasting the curve as a symmetric cover $X\to B$ and decomposing wild conductors via cyclic subcovers, together with an Artin-like induction across representations of $S_n$ and a perturbation-based local-constancy argument. A key outcome is that, under suitable ramification hypotheses, $n_{C,\text{wild}}$ equals the wild exponent of a hyperelliptic model $D$, i.e., $n_{C,\text{wild}}=n_{D,\text{wild}}$, with explicit formulas $n_{C,\text{wild}}=w_K(\text{disc}_x f)$ for plane curves $C: f(x,y)=0$ and $n_{C,\text{wild}}=w_K(g)$ for hyperelliptic $C: y^2=g(t)$ (when $p\neq2$). The results extend to superelliptic and more general covers, and an appendix corrects a 3-torsion computation in genus $2$. The methods provide a practical route to compute wild conductors (e.g., via Magma) and link non-hyperelliptic curves to hyperelliptic ones in terms of their wild ramification data.
Abstract
We give an explicit formula for wild conductor exponents of plane curves over $\mathbb{Q}_p$ in terms of standard invariants of explicit extensions of $\mathbb{Q}_p$. Furthermore, given a curve of genus $g$ with $p > g + 1$, we produce a hyperelliptic curve with the same wild conductor exponent. In an appendix we resolve a minor issue in the literature on the $3$-torsion of genus $2$ curves.