Wild conductor exponents of curves

1. Problem Statement

The paper addresses the problem of explicitly computing the wild conductor exponent ($n_{C, \text{wild}}$) of a curve $C$ defined over a finite extension $K$ of the $p$-adic numbers $\mathbb{Q}_p$.

• Context: The conductor $N = (\pi^{n_C})$ is a fundamental invariant measuring the bad reduction of an abelian variety (specifically the Jacobian $J_C$) at a prime $p$. It decomposes into a tame part ($n_{C, \text{tame}}$) and a wild part ($n_{C, \text{wild}}$).
• Difficulty: While the tame part is accessible via regular models, the wild part is generally difficult to determine. It depends on the action of the wild inertia subgroup of the absolute Galois group on the $\ell$-torsion of the Jacobian ($J_C[\ell]$) for $\ell \neq p$.
• Existing Solutions:
  • Tate's algorithm solves this for elliptic curves.
  • Explicit formulas exist for hyperelliptic curves (when $p \neq 2$) and specific low-genus cases (genus $\leq 5$).
  • Formulas exist for superelliptic curves under restrictive conditions ($p \nmid n$).
• Gap: There is no general, explicit formula for the wild conductor exponent of arbitrary plane curves or covers of $\mathbb{P}^1$ that relies on standard invariants of the defining polynomials, particularly when $p$ is large relative to the degree.

2. Methodology

Spencer employs a combination of representation theory, Galois module theory, and perturbation arguments to derive the results.

A. Motivic Pieces and Representation Theory

The author utilizes the framework of "motivic pieces" of curves (following Dokchitser et al.). For a curve $X$ with a $G$-action, the $\ell$-adic Tate module $V_\ell J_X$ decomposes into isotypic components $X_\rho = \text{Hom}_G(\rho, V_\ell J_X^*)$.

• The wild conductor exponent is additive over these components: $n_{X, \text{wild}} = \sum n_{X_\rho, \text{wild}}$.
• Key Lemma: For cyclic covers $C_n$, under the condition $p > \max \{q, (R_q-2)(q-1)+1\}$, the wild conductor exponent of any non-trivial irreducible representation $\rho$ equals that of the trivial representation ($n_{X_\rho, \text{wild}} = n_{X_1, \text{wild}}$). This effectively "kills" the contribution of non-trivial representations in the sum, simplifying the calculation.

B. Symmetric Covers and Induction

The paper generalizes the cyclic case to symmetric covers ($S_n$-covers).

• Theorem 5.2: Establishes a relation between the wild conductor of the standard representation $F$ of $S_n$ and the trivial/sign representations. Specifically, for a simply branched cover (with at most one exception), $n_{X_F, \text{wild}} = (n-2)n_{X_1, \text{wild}} + n_{X_\epsilon, \text{wild}}$.
• This allows the author to relate the conductor of a general degree $n$ cover to that of a degree 2 cover (hyperelliptic) sharing the same branch locus.

C. Perturbation and Local Constancy

To handle curves that are not naturally simply branched or to apply the results to general plane curves:

• Perturbation: The author shows that any smooth affine curve can be perturbed slightly (changing coefficients $p$-adically) to become a simply branched cover of $\mathbb{P}^1$ without changing the genus or the wild conductor exponent.
• Local Constancy (Lemma 6.3): The quantity $w_K(g)$, defined as a weighted sum of valuations of discriminants of roots of a polynomial $g$, is locally constant under small $p$-adic perturbations of the coefficients of $g$.
• Reduction: By perturbing a general curve $C: f(x,y)=0$ to a simply branched cover, the problem reduces to computing the conductor of an associated hyperelliptic curve defined by the discriminant of the projection.

3. Key Contributions and Results

Main Theorem (Theorem 1.1 & 5.2)

For a curve $C/K$ equipped with a simply branched degree $n$ cover of $\mathbb{P}^1$ (where $p > n$), the wild conductor exponent is equal to that of any hyperelliptic curve $D$ whose degree 2 map to $\mathbb{P}^1$ has the same branch locus as the cover of $C$.
$$n_{C, \text{wild}} = n_{D, \text{wild}}$$
This generalizes the known results for hyperelliptic curves to a much broader class of curves.

Explicit Formula for Plane Curves (Theorem 1.4 / Theorem 6.5)

For a smooth affine curve $C: f(x, y) = 0$ over $K$ with smooth compactification, if $p > \deg_x f$, the wild conductor exponent is given by:
$$n_{C, \text{wild}} = w_K(\text{disc}_x f)$$
Where $w_K(g)$ is defined as:
$$w_K(g) = \sum_{r \in R/G_K} m(r) \cdot (v_K(\Delta_{K(r)/K}) - [K(r):K] + f_{K(r)/K})$$
Here, $R$ is the set of roots of $g$, $m(r)$ is multiplicity, $\Delta$ is the relative discriminant, and $f$ is the residue degree.

Application to Superelliptic Curves (Corollary 1.7)

For a superelliptic curve $C: y^n = f(x)$ with $f$ square-free and $p > n$:
$$n_{C, \text{wild}} = (n-1) \cdot w_K(f)$$
This recovers and generalizes previous formulas for superelliptic curves.

Correction to Literature (Appendix A)

The author identifies and fixes a minor error in [7, Proposition 4.1] regarding the correspondence between non-zero 3-torsion points of genus 2 curves and specific polynomial tuples.

• Issue: The original proposition failed to account for cases where the curve admits a degree 3 map to $\mathbb{P}^1$ that is not simply branched (specifically, when the associated discriminant curve has genus $< 2$).
• Fix: The corrected proposition (Proposition A.2) includes three distinct sets of tuples to cover all cases, ensuring a one-to-one correspondence with the 3-torsion points.

4. Significance and Impact

1. Algorithmic Computation: The paper provides a practical, explicit formula for computing wild conductor exponents for a wide class of curves (plane curves, superelliptic curves) using standard polynomial invariants (discriminants). This is implementable in computer algebra systems like Magma (as demonstrated in Example 1.9).
2. Generalization: It unifies the treatment of conductors for hyperelliptic and superelliptic curves under a single framework based on the branch locus of the cover, removing the need for case-specific ad-hoc methods.
3. Theoretical Insight: The work clarifies the relationship between the wild inertia action on the Jacobian of a general cover and the roots of the discriminant polynomial. It demonstrates that, under the condition $p > n$, the complex structure of the Jacobian's torsion is governed entirely by the ramification data of the cover.
4. Correction: The appendix ensures the accuracy of 3-torsion computations for genus 2 curves, which is critical for algorithms relying on these structures (e.g., in the study of L-functions or BSD conjectures).

In summary, Spencer's work bridges the gap between abstract Galois module theory and explicit arithmetic geometry, offering a robust tool for determining the bad reduction of curves over $p$-adic fields.