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Ekedahl-Oort Types and Newton Polygons of Abelian Covers of $\mathbf{P}^1$ Branched at Three Points

This paper investigates the Newton polygons and Ekedahl-Oort types of abelian covers of the projective line branched at three points, demonstrating that supersingular and superspecial curves, as well as "unlikely" geometric properties, occur with a higher frequency than expected and providing new evidence for Oort's Conjecture.

Original authors: Darren Schmidt

Published 2026-02-10

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CC BY 4.0

Original authors: Darren Schmidt

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a mathematician looking at a vast, infinite landscape of shapes called "curves." In the world of geometry, these curves have "DNA"—special mathematical fingerprints called Newton Polygons and Ekedahl-Oort types. These fingerprints tell us how the curve behaves when we look at it through a special lens (called "characteristic $p$ ").

Most curves in this landscape follow predictable patterns. However, every once in a while, you find a "glitch in the Matrix"—a curve with a fingerprint so rare and strange that, according to the standard rules of probability, it shouldn't really exist.

This paper, written by Darren Schmidt, is essentially a study of these "mathematical glitches."

1. The "Unlikely" Glitches

Think of the "moduli space" (the landscape of all possible curves) like a giant city. Most buildings in this city are standard apartments. A "supersingular" or "superspecial" curve is like finding a solid gold skyscraper in a neighborhood of wooden shacks.

According to the "naive" rules of math, these golden skyscrapers should be incredibly rare—so rare that you might never find one as the city gets bigger. But Schmidt studied a specific family of curves (called abelian covers of the projective line) and discovered something shocking: The golden skyscrapers are everywhere.

He found that these "unlikely" curves appear much more frequently than anyone expected. It’s like walking into a desert and finding that instead of a few rare oases, there are actually massive, lush forests hiding just beneath the surface.

2. The "Recipe" for Glitches (Theorem 1.6)

Schmidt didn't just observe these glitches; he figured out the recipe to bake them.

Imagine you are a chef trying to create a very specific, rare flavor of cake. Usually, you’d have to guess ingredients blindly. Schmidt discovered a mathematical "formula" (Theorem 1.6). He showed that if you pick certain prime numbers (the "ingredients") and follow a specific structural pattern, you are guaranteed to produce these rare, "unlikely" curves.

He proved that you can create an infinite number of these "glitches," which challenges the old assumption that they were just rare accidents.

3. The "Large Denominator" Mystery

There is another way a curve can be "weird." Sometimes, the "DNA" of a curve has numbers that are incredibly complex (large denominators). In the math world, this is like finding a person whose DNA is written in a language so complex it shouldn't be possible for a human to speak it.

Schmidt used his new "recipe" to prove that these hyper-complex curves also exist in infinite numbers, even when they are so "unlikely" that they defy previous mathematical expectations.

4. The Big Picture: Why does this matter?

In mathematics, we often rely on "heuristics"—educated guesses about how things should behave. For a long time, mathematicians thought the landscape of curves was "transverse," meaning the rare glitches and the common curves didn't overlap much.

Schmidt’s work shows that the landscape is much more "tangled" than we thought. The rare and the common are deeply intertwined. By proving that these glitches appear with high density (sometimes appearing in more than 99% of cases!), he is helping to rewrite the map of the mathematical universe.

In short: Schmidt found that the "impossible" is actually quite common, and he gave us the manual to build it ourselves.

Technical Summary: Ekedahl-Oort Types and Newton Polygons of Abelian Covers of $\mathbb{P}^1$ Branched at Three Points

Author: Darren Schmidt

1. Problem Statement

The paper investigates the distribution of specific invariants of the Jacobians of smooth projective curves in positive characteristic $p$ . Specifically, it focuses on Newton polygons (which describe the $p$ -adic properties of the Jacobian) and Ekedahl-Oort (EO) types (which describe the isomorphism class of the $p$ -torsion group scheme).

A central question in arithmetic geometry is determining which Newton polygons or EO types are "realized" by the Jacobians of smooth curves. The author focuses on a specific family: abelian covers of the projective line $\mathbb{P}^1$ branched at exactly three points. These curves are significant because their Jacobians possess Complex Multiplication (CM), which allows for a more tractable analysis of the Frobenius and Verschiebung operators via the Shimura-Taniyama formula.

The paper specifically addresses the phenomenon of "unlikely" invariants. An invariant is "unlikely" if the codimension of its corresponding stratum in the moduli space of abelian varieties ( $\mathcal{A}_g$ ) exceeds the dimension of the moduli space of curves ( $\mathcal{M}_g$ ). The author seeks to determine how frequently these unlikely invariants occur in this specific family of curves.

2. Methodology

The author employs a combination of computational and theoretical approaches:

Computational (SageMath): The author developed a new algorithm in SageMath capable of handling non-cyclic abelian covers. This allows for the computation of Newton polygons and EO types for genus $g$ up to 50. By analyzing the congruence conditions on the prime $p$ that yield specific curve types, the author calculates the natural density of such primes.
Theoretical (Shimura-Taniyama Formula): To move beyond computation, the author uses the Shimura-Taniyama formula to relate the slopes of the Newton polygon to the distribution of quadratic residues and non-residues in specific intervals modulo a prime $\ell$ .
Algebraic Construction: The author constructs specific cyclic covers using a parameter $\ell$ (a prime) and $n$ (an integer) to create curves with controlled ramification and inertia types. This construction is used to prove existence theorems for supersingular curves and unlikely Newton polygons.

3. Key Contributions and Results

A. Computational Findings on Densities:
The author provides empirical evidence that "unlikely" invariants are far more common than a naive dimensional heuristic would suggest. For $9 \leq g \leq 50$ , the natural densities ( $\delta$ ) of primes yielding specific curves are remarkably high:

Supersingular curves ( $\delta_{ss}$ ): $> 0.7$
Superspecial curves ( $\delta_{ssp}$ ): $> 0.2$
Unlikely Newton polygons ( $\delta_{nu}$ ): $> 0.875$

B. Theoretical Breakthroughs:

Theorem 1.6 (Existence Theorem): The author proves a theorem providing a construction for cyclic covers where the Newton polygon slopes are explicitly determined by the number of quadratic residues in specific intervals modulo $\ell$ . This provides infinitely many examples of supersingular curves.
Refutation of Expectation 1.5: The author provides a counterexample to an expectation by Oort, showing that Newton polygons with "large denominators" (which were expected not to be realized for all primes) can indeed be realized in certain odd characteristics.
Oort’s Conjecture: The results provide evidence for Oort’s conjecture regarding the realization of the direct sum of Newton polygons ( $\xi_1 \oplus \xi_2$ ).

C. Asymptotic Behavior and Conjectures:

Conjecture 1.3: Based on the data, the author conjectures that for sufficiently large $g$ , almost all primes (except those where $p \equiv 1 \pmod m$ ) yield curves with unlikely Newton polygons or EO types.
Theorem 1.12: The author proves that if Dickson’s Conjecture (on the existence of prime constellations) is true, then the natural density of primes yielding supersingular curves approaches 1 as the genus $g \to \infty$ .

4. Significance

This paper is significant for several reasons:

Correcting the Heuristic: It demonstrates that the "dimensional transversality" heuristic—the idea that unlikely strata should not intersect the Torelli locus—is fundamentally flawed in the context of curve moduli.
Algorithmic Advancement: The extension of computational tools to non-cyclic covers allows for the exploration of much larger genus ranges than previous studies.
Bridging Theory and Computation: By using computational data to form conjectures and then using rigorous number theory (quadratic excess, Shimura-Taniyama) to prove them, the paper provides a robust framework for studying the arithmetic of curves in characteristic $p$ .
New Constructions: It provides explicit, new methods for constructing supersingular curves and curves with specific, "unlikely" Newton polygons, which are of interest to researchers in arithmetic geometry and coding theory.