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arxiv: 2604.20091 · v1 · submitted 2026-04-22 · 🧮 math.NT · math.AG

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

Bryden Cais , Douglas Ulmer This is my paper

Pith reviewed 2026-05-09 23:52 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords Artin-Schreier coversa-numberordinary curvesZariski open setscharacteristic palgebraic curvespolynomial covers
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0 comments X

The pith

Generic Artin-Schreier curves y^p - y = f(x) attain the established lower bound on a-numbers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a non-empty Zariski open subset U inside the space of degree-d polynomials where d is not divisible by the characteristic p. For every polynomial f in this open set over a perfect field k, the a-number of the curve y^p - y = f(x) equals a fixed lower bound value. This shows the bound is achieved for a dense collection of such covers and is therefore tight. The same computation implies that the minimal a-numbers of Artin-Schreier covers of ordinary curves are likewise attained on an open set.

Core claim

The central claim is that there exists a non-empty Zariski open subset U of the space of degree-d polynomials such that for every f in U(k), the a-number of the Artin-Schreier curve y^p - y = f(x) equals the lower bound value. This uniform computation on U demonstrates that the bound is attained and hence sharp, and it further shows that the minimal a-numbers for Artin-Schreier covers of ordinary curves equal this same value.

What carries the argument

The non-empty Zariski open subset U of the space of degree-d polynomials, on which the a-number of the Artin-Schreier curve is constant and equal to the lower bound.

If this is right

  • The lower bound for a-numbers of Artin-Schreier covers is attained for a dense open set of polynomials.
  • The bound on minimal a-numbers for Artin-Schreier covers of ordinary curves is tight.
  • Most Artin-Schreier covers of ordinary curves realize the smallest possible a-number in their class.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result indicates that minimal a-numbers occur densely in the parameter space of these covers.
  • The same open-set technique could be used to compute other p-adic invariants such as Newton polygons for generic Artin-Schreier curves.
  • Explicit checks for small values of p and d could verify the density statement in concrete cases.

Load-bearing premise

There exists a non-empty Zariski open subset U of the space of degree-d polynomials such that the a-number equals the lower bound for all f in U(k) and the resulting curves are ordinary when required.

What would settle it

An explicit polynomial f of degree d not divisible by p, lying outside any exceptional locus, for which the a-number of y^p - y = f(x) strictly exceeds the lower bound value would show the claim is false.

Figures

Figures reproduced from arXiv: 2604.20091 by Bryden Cais, Douglas Ulmer.

Figure 1
Figure 1. Figure 1: An index set for differentials It is clear that H ≤J = ⊕J ≤JH J and H J = pH J ⊕ ′H J . The following subspaces will be useful in considering the range of C. Definition 2.3. For 0 < ℓ < p and ℓ ≤ e < p, let H(ℓ, e) := ωi,j [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

Let $k$ be a perfect field of characteristic $p>0$, and let $d$ be a positive integer not divisible by $p$. We define a non-empty Zariski open subset $U$ of the space of polynomials of degree $d$, and for $f(x)\in U(k)$, we compute the $a$-number of the curve defined by $y^p-y=f(x)$. This $a$-number realizes a lower bound given by Booher and Cais, so the latter is tight. Our result also implies that the bound of Booher and Cais for minimal $a$-numbers of Artin-Schreier covers of ordinary curves is tight.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript defines a non-empty Zariski open subset U of the space of degree-d polynomials (d not divisible by p) over a perfect field k of characteristic p. For f in U(k), it computes the a-number of the Artin-Schreier curve y^p - y = f(x) and shows that this value equals the lower bound previously obtained by Booher and Cais, thereby establishing that the bound is attained. The authors further conclude that their result implies the Booher-Cais bound is tight when restricted to Artin-Schreier covers of ordinary curves.

Significance. If the central computation holds and the ordinary-curve implication is valid, the paper supplies an explicit, Zariski-dense family of Artin-Schreier covers realizing the minimal a-number, confirming the sharpness of the Booher-Cais bound in both the general and ordinary settings. This would be a concrete advance in the study of p-torsion invariants of Jacobians of curves in characteristic p.

major comments (1)
  1. [statement of the main result (immediately after the definition of U)] The implication that the Booher-Cais bound is tight for ordinary Artin-Schreier covers requires that U intersects the ordinary locus non-emptily (i.e., that there exist f in U(k) for which the curve y^p - y = f(x) has p-rank equal to its genus). The manuscript provides no explicit verification or argument that the coefficient conditions defining U are compatible with ordinariness; if the generic member of U has positive a-number forcing reduced p-rank, the claim for ordinary curves does not follow from the computation on U.
minor comments (1)
  1. [abstract] The abstract states that the a-number 'realizes a lower bound' but does not record the explicit value obtained; including the formula would make the main theorem easier to locate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their careful reading and for highlighting an important point regarding the implication for ordinary curves. We address the major comment below and outline the changes we will make.

read point-by-point responses

  1. Referee: [statement of the main result (immediately after the definition of U)] The implication that the Booher-Cais bound is tight for ordinary Artin-Schreier covers requires that U intersects the ordinary locus non-emptily (i.e., that there exist f in U(k) for which the curve y^p - y = f(x) has p-rank equal to its genus). The manuscript provides no explicit verification or argument that the coefficient conditions defining U are compatible with ordinariness; if the generic member of U has positive a-number forcing reduced p-rank, the claim for ordinary curves does not follow from the computation on U.

    Authors: We agree with the referee that the manuscript does not provide an explicit argument showing that U intersects the ordinary locus non-emptily, and that such an intersection is required to support the claim about ordinary Artin-Schreier covers. Upon reflection, since the a-number realized on U equals the Booher-Cais lower bound (which is positive for the degrees under consideration), and given the relation a + r ≤ g between the a-number and p-rank r, the curves for generic f in U necessarily have r < g and thus lie outside the ordinary locus. We will therefore revise the manuscript by removing the sentence claiming that the result implies tightness of the Booher-Cais bound for ordinary curves. The core result—that the lower bound is achieved for a Zariski-dense set of Artin-Schreier curves—remains valid and is unaffected by this change. revision: yes

Circularity Check

0 steps flagged

No significant circularity; independent explicit computation achieves prior lower bound

full rationale

The paper defines a non-empty Zariski open subset U of the space of degree-d polynomials and performs an explicit computation of the a-number for the Artin-Schreier curve y^p - y = f(x) when f lies in U(k). This value is shown to equal the lower bound previously established by Booher and Cais, thereby demonstrating that the bound is attained. The derivation consists of a direct calculation on the generic locus U and does not reduce the a-number to any fitted parameter, self-referential definition, or ansatz imported from the cited work. The self-citation supplies only the independent lower bound being matched; the matching itself is obtained from the paper's own equations and does not rely on a chain that collapses back to the input. While the further implication for ordinary curves requires that generic members of U be ordinary, this is a separate question of whether the constructed examples lie in the ordinary locus and does not render the core computation circular or self-referential.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of a-numbers, Artin-Schreier covers, and Zariski topology over perfect fields of characteristic p; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard properties of the a-number for Jacobians of curves in characteristic p and of Artin-Schreier extensions.
    Invoked implicitly when stating the lower bound and the computation for the curve y^p - y = f(x).

pith-pipeline@v0.9.0 · 5410 in / 1216 out tokens · 38617 ms · 2026-05-09T23:52:10.270186+00:00 · methodology

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