The minimum genus of Galois covers of curves

Technical Summary: The Minimum Genus of Galois Covers of Curves

Problem Statement
The paper addresses the problem of determining the minimum genus of Galois covers of smooth projective connected curves over an algebraically closed field $k$ of positive characteristic $p$. Specifically, the authors investigate the existence and minimal genus of $\Gamma$-covers $W \to X$ that are étale outside a finite set of closed points $S_X \subset X$, where $\Gamma$ is a finite group extension of an elementary abelian $l$-group $H \cong (\mathbb{Z}/l\mathbb{Z})^n$ by a cyclic $p$-group $G \cong \mathbb{Z}/p^a\mathbb{Z}$.

The study is motivated by the structure of the étale fundamental group $\pi_1^{\text{ét}}(X \setminus S_X)$. While Abhyankar's conjecture (proved by Raynaud and Harbater) characterizes which finite groups appear as quotients of this fundamental group, it does not fully describe the inverse system of finite quotients. Given a $G$-cover $V \to X$, the authors seek to characterize the $\Gamma$-covers $W \to X$ that dominate $V$ and determine the minimum possible genus for such covers. This extends previous work on covers with Galois groups isomorphic to alternating groups or specific semi-direct products where the action was fixed or restricted.

Methodology
The authors employ a combination of algebraic geometry, representation theory, and combinatorial optimization:

1. Module Structure of Torsion Points: The core of the methodology relies on analyzing the $l$-torsion subgroup of the Picard group of the intermediate cover $V$, denoted $Pl(V \setminus S_V)$, where $S_V$ is the inverse image of $S_X$. This group is a right $G$-module. The authors utilize the decomposition of this module into irreducible $G$-representations over $\mathbb{F}_l$.
2. Galois Correspondence: Building on prior results (specifically [13]), the paper establishes a bijection between $H$-covers of $V$ (which form $\Gamma$-covers of $X$) and $G$-stable submodules of $Pl(V \setminus S_V)$ isomorphic to $H$.
3. Decomposition Theorems: The authors derive explicit decompositions of $Pl(V \setminus S_V)$ into irreducible components. They distinguish between the part coming from the degree-zero Picard group, $Pic^0(V)[l]$, and the part arising from the ramification divisor, $Pram(S_V, l)$. The structure of $Pram(S_V, l)$ is determined by the stratification of the branch locus $S_X$ based on the ramification indices of the $G$-cover $V \to X$.
4. Optimization via Linear Programming: To find the minimum genus, the problem is reduced to an integer linear programming problem. The genus of the cover $W$ is a function of the number of ramification points in the $H$-cover $W \to V$. Minimizing the genus corresponds to minimizing the weighted sum of ramification points, subject to constraints imposed by the multiplicities of irreducible representations in the module $H/H'$, where $H'$ is the maximal submodule of $H$ embeddable in $Pic^0(V)[l]$.

Key Contributions and Results

• General Formula for Minimum Genus (Theorem 1.1): The paper provides a method to calculate the minimum genus $g$ of a $\Gamma$-cover dominating a fixed $G$-cover $V$. This is formulated as an integer linear programming problem:

  • Objective: Minimize $\sum_{s=0}^a p^s y_s$, where $y_s$ represents the number of branch points in $S_X$ with specific ramification properties.
  • Constraints: The variables $y_s$ are bounded by the number of available points in $S_X$ and must satisfy inequalities derived from the multiplicities of irreducible $G$-modules ($M_{s,t}$) in the quotient $H/H'$.
  • Genus Formula: Once the minimum value $c_{\min}$ of the objective function is found, the minimum genus is given by:
    $$g = 2 + l^n(2g_V - 2) + \frac{l^{n-1}(l-1)c_{\min}}{2}$$
    This generalizes previous results by accounting for arbitrary actions $\theta: G \to \text{Aut}(H)$ and varying ramification structures of the base cover $V$.

• Specific Cases and Explicit Formulas:

  • Theorem 4.2: Specializes the general result to cases where the branch locus consists entirely of totally ramified points or points with exactly $p$ pre-images, providing explicit formulas for the constant $c$ in the genus equation.
  • Theorem 5.6 and Corollary 5.7: Focus on the case where $X = \mathbb{P}^1$ and the cover is étale over $\mathbb{A}^1$ (branched only at infinity). The authors define upper jumps in the ramification filtration and relate them to the decomposition of the module $H$. They provide a recursive method to compute the minimum genus based on the "upper jumps" $u_i$.

• Minimum Genus for Varying Actions (Theorem 1.2): A significant contribution is the calculation of the minimum genus over all possible group actions $\theta: \mathbb{Z}/p^2\mathbb{Z} \to \text{Aut}((\mathbb{Z}/l\mathbb{Z})^n)$ and all valid $n$. For $a=2$, the authors derive a closed-form expression for the minimum genus depending on the order of $l$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ and $(\mathbb{Z}/p^2\mathbb{Z})^\times$.

  • If $d_1 = d_2$ (orders are equal), the minimum is achieved when $n=d_1$.
  • If $d_1 < d_2$, the minimum depends on whether $p=2$ or $p \neq 2$, yielding specific constants ($15$ for $p=2$, $2p^2+2$ for $p \neq 2$).

Significance and Claims
The paper claims to provide a systematic and algorithmic approach to solving the "embedding problem" for Galois covers with a specific semi-direct product structure. By decomposing the $l$-torsion of the Picard group into irreducible $G$-modules, the authors translate a geometric problem (finding covers with minimal genus) into a combinatorial optimization problem.

The significance lies in:

1. Generalization: It extends known results on minimal genus covers (such as those in [5] for $a=1$) to the case of higher $p$-power cyclic groups ($a \geq 2$) and arbitrary actions.
2. Completeness: It addresses the scenario where the group action $\theta$ is not fixed but varies, determining the absolute minimum genus achievable within the class of groups $(\mathbb{Z}/l\mathbb{Z})^n \rtimes \mathbb{Z}/p^a\mathbb{Z}$.
3. Explicit Computation: The paper offers concrete formulas and linear programming constraints that allow for the explicit calculation of these minimum genera, rather than just proving existence.

The authors note that their results recover earlier findings (e.g., [5, Theorem 4.2 and 4.3]) as special cases and provide a framework for understanding the inverse system of finite quotients of the étale fundamental group of affine curves in positive characteristic.