Minimal $a$-numbers of Artin--Schreier covers of ordinary curves

1. Problem Statement and Context

The Core Problem:
The paper addresses the computation of the $a$-number for Artin–Schreier covers of curves in characteristic $p > 0$.

• Context: Let $k$ be a perfect field of characteristic $p$. Let $X = \mathbb{P}^1$ and consider the Artin–Schreier cover $Y$ defined by the equation $y^p - y = f(x)$, where $f(x)$ is a polynomial of degree $d$ (with $p \nmid d$).
• The $a$-number: For a curve $Z$, the $a$-number is defined as $a_Z = \dim_k \ker(C)$, where $C$ is the Cartier operator acting on the space of regular differentials $H^0(Z, \Omega^1_Z)$. It measures the complexity of the $p$-torsion of the Jacobian $J_Z[p]$. Specifically, $0 \leq a_Z \leq g_Z$.
• The Gap: Previous work by Booher and Cais ([BC20]) established lower and upper bounds for the $a$-number of such covers. While these bounds coincided (and were sharp) for $p=2$, and optimality was proven for $p=3, 5$ and specific examples for other $p$, a general proof that the lower bound is tight for all $p$ and all degrees $d$ was missing.

The Goal:
To prove that for a generic polynomial $f(x)$ of degree $d$, the $a$-number of the resulting curve $Y$ achieves the theoretical lower bound proposed by Booher and Cais. Furthermore, the authors aim to extend this result to Artin–Schreier covers of general ordinary curves (curves with $a_X = 0$).

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2. Methodology

The authors employ a combination of filtration techniques, linear algebra over function fields, and algebraic geometry (Zariski topology).

A. Filtration of the Space of Differentials

The space of regular differentials $H = H^0(Y, \Omega^1_Y)$ has a basis indexed by pairs $(i, j)$:
$$\omega_{i,j} = y^{j-1} x^{i-1} dx$$
where $0 < j < p$ and $0 < i < \frac{(p-j)d}{p}$.
The authors define a filtration on $H$ based on the index $j$:
$$H_{\leq J} = \langle \omega_{i,j} \mid 0 < j \leq J \rangle$$
They further decompose the "level" spaces $H_J$ (where $j=J$) into subspaces based on divisibility of $i$ by $p$:

• $pH_J$: spanned by $\omega_{i,J}$ where $p \mid i$.
• $'H_J$: spanned by $\omega_{i,J}$ where $p \nmid i$.

B. Analysis of the Cartier Operator

The core of the proof involves analyzing the action of the Cartier operator $C$ on these subspaces.

• Proposition 2.6: The authors derive explicit formulas for $C(\omega_{i,j})$. They show that $C$ maps $H_J$ into a direct sum of specific subspaces $H(\ell, J)$ (defined by degree constraints on the resulting polynomials).
• Key Map $\Phi_J$: They define a map $\Phi_J$ which is the composition of $C$ restricted to $'H_J$ followed by a projection onto the "quotient" of the target spaces (removing the "top layer" corresponding to $pH_J$).
  $$\Phi_J: 'H_J \to \bigoplus_{\ell=1}^{J-1} \frac{H(\ell, J)}{H(\ell, J-1)}$$

C. Genericity and Determinantal Arguments

To prove the result for a "generic" polynomial, the authors treat the coefficients of $f(x)$ as variables in a polynomial ring $S = k[a_0, \dots, a_d]$.

1. Matrix Representation: They construct a matrix representing the map $\Phi_J$ in terms of the coefficients of $f(x)$. The entries of this matrix are coefficients of powers of the generic polynomial $f$.
2. Rank Analysis: The goal is to show that for a generic choice of coefficients, this matrix has maximal rank (i.e., it is surjective or injective depending on the range of $J$).
3. Non-vanishing Determinant: They identify a specific maximal minor of this matrix. By analyzing the leading monomials of the entries (using a specific monomial ordering and weight grading), they prove that the determinant of this minor is a non-zero polynomial in the coefficients.
4. Zariski Open Set: Since the determinant is non-zero, the set of coefficients where the matrix fails to have full rank is a proper Zariski closed subset. Thus, there exists a non-empty Zariski open subset $U$ of the space of polynomials where the map has maximal rank.

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3. Key Contributions and Results

Theorem 1.3 (Main Result)

There exists a non-empty Zariski open subset $U$ of the space of polynomials of degree $d$ such that for any $f \in U(k)$, the $a$-number of the curve $y^p - y = f(x)$ is exactly:
$$L(d) = \sum_{\ell=1}^{(p-1)/2} \left( \left\lfloor \frac{(p-\ell)d}{p} \right\rfloor - \left\lfloor \frac{(p^2 + 1 - 2\ell p)d}{2p^2} \right\rfloor \right)$$
This confirms that the Booher–Cais lower bound is tight for all $p$ and all $d$.

Theorem 2.5 (Refined Structural Result)

The paper provides a refined description of the kernel of the Cartier operator:

• For $J \leq (p+1)/2$, the restriction of $C$ to $H_{\leq J}$ induces a surjection onto a specific direct sum of subspaces.
• The kernel of $C$ on the entire space $H$ is contained entirely within $H_{\leq (p-1)/2}$.
  This structural insight allows for the precise calculation of the dimension of the kernel (the $a$-number).

Corollary 1.4 (Generalization to Ordinary Curves)

The result extends to Artin–Schreier covers of any smooth, irreducible, projective ordinary curve $X$ (where $a_X = 0$).
If $\pi: Y \to X$ is a cover ramified at points $P_1, \dots, P_B$ with ramification invariants $d_1, \dots, d_B$ (all prime to $p$), then for a generic choice of the defining function $f$, the $a$-number of $Y$ is the sum of the minimal $a$-numbers for each local ramification:
$$a_Y = \sum_{b=1}^B L(d_b)$$

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4. Significance

1. Resolution of a Conjecture: The paper settles the question of whether the Booher–Cais lower bound for minimal $a$-numbers is optimal. It proves that for every prime $p$ and every degree $d$, there exist curves achieving this minimum.
2. Explicit Construction: Unlike previous results that relied on specific examples or small primes, this paper provides a constructive proof that such curves exist generically. It defines a specific open set $U$ where the minimal $a$-number is realized.
3. Methodological Innovation: The technique of using a filtration on the space of differentials combined with a detailed analysis of the leading monomials of the Cartier operator's matrix representation is a powerful tool. It demonstrates how to handle the non-linearity of the Cartier operator in a generic setting.
4. Implications for Jacobians: Since the $a$-number determines the structure of the $p$-torsion of the Jacobian, these results provide a deeper understanding of the distribution of $p$-torsion group schemes in families of Artin–Schreier curves. It shows that the "most ordinary" behavior (minimal $a$-number) is the generic behavior for these covers.

In summary, Cais and Ulmer provide a rigorous proof that the theoretical lower bound for the $a$-number of Artin–Schreier covers is always achievable, utilizing advanced filtration techniques and algebraic geometry to establish the genericity of this minimal case.