Overview
Research published on arXiv presents conductor-discriminant inequalities for a specific class of algebraic curves. The focus is on $\mathbb{Z}/n$-covers of the projective line $\mathbb{P}^1$. These curves are considered over discretely valued fields, denoted $K$, which possess an excellent valuation ring, $\mathcal{O}_K$, and a perfect residue field. A crucial condition for the residue field is that its characteristic must not divide $n$. The derived inequalities are contingent upon calculations detailed in a separate work by the third author (arXiv:2609.20585).
Approach
The study investigates curves $X$ expressed in the form $y^n = f(x)$, where $f(x)$ is a polynomial with coefficients in $\mathcal{O}_K[x]$ and the degree of $f$ is a multiple of $n$, i.e., $n\mid\text{deg}(f)$. For such a curve $X$, its minimal regular model over $\mathcal{O}_K$ is denoted $\mathcal{X}$. The research provides a specific inequality involving the Artin conductor of this minimal regular model.
Findings
The core finding establishes an upper bound for the negative of the Artin conductor of $\mathcal{X}$. Specifically, the inequality states that the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$. Here, $v_K$ represents the valuation in the field $K$, and $\text{disc}(\text{rad}(f))$ refers to the discriminant of the radical of the polynomial $f(x)$.
- The derived inequality is applicable to $\mathbb{Z}/n$-covers of $\mathbb{P}^1$ defined over discretely valued fields $K$.
- The fields $K$ must have an excellent valuation ring $\mathcal{O}_K$ and a perfect residue field.
- A condition on the residue field is that its characteristic must not divide $n$.
- The result holds for curves $X$ given by $y^n = f(x)$, where $f(x) \in\mathcal{O}_K[x]$ and $n\mid\text{deg}(f)$.
- The negative of the Artin conductor of the minimal regular model $\mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$.
Why This Matters
This research represents a direct generalization of earlier work concerning hyperelliptic curves, conducted by the first two authors of the current study. This previous work had, in turn, extended results established by Ogg, Saito, Liu, and the second author. Furthermore, when the polynomial $f$ is monic, the present findings strengthen a result previously put forth by Kohls. Kohls' work indicated that the conductor exponent of the Jacobian of such a curve is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$. The current research provides a more stringent bound under specific conditions.