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Superelliptic curves with large Galois images

This paper establishes conditions under which the mod $\ell$ Galois representations of superelliptic curves $y^r = f(x)$ over cyclotomic fields have large images, providing the first explicit results for abelian varieties of dimension greater than two that are not of ${\rm GL}_2$ -type and allowing for unramified extensions of the ground field.

Original authors: Pip Goodman

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pip Goodman

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 detective trying to solve a massive, cosmic puzzle. Your job is to figure out how a specific set of "keys" (mathematical numbers) can unlock a vast, hidden "vault" (a mathematical structure called a Galois representation).

This paper, written by Pip Goodman, is about finding the perfect keys to unlock these vaults for a special type of mathematical curve called a superelliptic curve.

Here is the breakdown of the adventure, using simple analogies:

1. The Setting: The Curious Curves

Think of a superelliptic curve as a twisted, multi-layered rope. It's defined by an equation like $y^r = f(x)$ .

$x$ and $y$ are the coordinates on the rope.
$r$ is how many times the rope twists around itself.
$f(x)$ is the shape of the rope's path.

The author is interested in what happens when you look at these curves through a special lens called mod $\ell$ . This lens breaks the infinite complexity of the curve down into a finite, manageable grid of points (like looking at a high-resolution photo and zooming in until you see individual pixels).

2. The Goal: The "Large" Image

When you look at these curves through the lens, the points move around in a very specific way. Mathematicians call this movement a Galois representation.

Imagine the points are dancers on a stage.
The Galois group is the choreographer telling them how to move.
The Image is the pattern the dancers make.

Usually, the dancers might only move in a small, restricted circle (a "small image"). But the author wants to prove that for certain curves, the dancers can perform every possible dance move allowed by the rules of the stage. This is called a "large image."

Why does this matter? If the image is "large," it means the curve is "wild" and unpredictable in a very structured way. This helps mathematicians solve the Inverse Galois Problem, which is essentially asking: "Can we build a specific type of lock (a mathematical group) using a specific type of key (a curve)?"

3. The Challenge: The "Endomorphism" Trap

There's a catch. These curves have a built-in symmetry (like a snowflake that looks the same if you rotate it). This symmetry acts like a guard that restricts the dancers.

In simpler curves (like elliptic curves), the guard is weak, and the dancers can do almost anything.
In these complex superelliptic curves, the guard is strong. It forces the dancers to stay in a specific formation.

The author's main job is to prove that even with this strong guard, the dancers can still perform almost every possible move (specifically, they can form the largest possible "perfect" subgroup of the allowed moves).

4. The Tools: The "Endomorphism Character"

To prove the dancers can move freely, the author invents a special tool called the "Endomorphism Character."

Analogy: Imagine the guard has a secret code book. The author finds a way to read this code book.
This "character" is a mathematical fingerprint that tells the author exactly how the symmetry guard restricts the dancers.
By understanding this fingerprint, the author can predict exactly which dance moves are possible and prove that the "forbidden" moves are actually just a tiny, manageable list of exceptions.

5. The Strategy: The "Newton Polygon" Map

How does the author find the perfect curves? They use a map called a Newton Polygon.

Analogy: Imagine you are building a bridge. You need to know exactly where the rocks (coefficients of the equation) are placed so the bridge doesn't collapse.
The Newton Polygon is a blueprint that shows the author where to place the numbers in the equation $f(x)$ so that the curve behaves exactly as needed.
The author shows that if you place the numbers in a specific "staircase" pattern (using prime numbers), the curve will have the "large image" property.

6. The Results: Building the Bridges

The paper doesn't just talk theory; it builds actual examples.

The author constructs specific curves with degrees (sizes) ranging from 12 to 30.
For example, they show a curve where $y^3 = x^{12} + \dots$ has a "large image" for almost all prime numbers $\ell$ .
They even provide a "cheat sheet" (a list of specific primes to avoid) so that if you pick any other prime, the math works perfectly.

7. Why This is a Big Deal

New Territory: Before this, mathematicians mostly understood these "large image" properties for simple curves (like 2D surfaces). This paper breaks into higher dimensions (curves that are much more complex, like 10D or 30D objects).
Unrestricted Ground: Previous methods required the "ground" (the number field) to be very simple. This paper works on more complex grounds (fields with "unramified extensions"), which is like building a skyscraper on a swampy, unstable foundation. The author proves it's possible.
First Time: For the case where $r=3$ , this is the first time anyone has accurately described the entire set of possible dance moves for these curves.

Summary

Pip Goodman has written a guidebook for building complex mathematical curves that are as "wild" and "free" as possible, despite having built-in symmetries that try to hold them back. By using a new "fingerprint" tool and a clever "blueprint" for placing numbers, the author proves that these curves can unlock the most complex mathematical vaults, solving a piece of a century-old puzzle about how numbers and shapes interact.

1. Problem Statement

The paper addresses the problem of determining the image of mod $\ell$ Galois representations attached to the Jacobians of superelliptic curves of the form:
$C: y^r = f(x)$
where $r > 2$ is a prime, $f(x)$ is a monic polynomial with coefficients in the $r$ -th cyclotomic field $\mathbb{Q}(\zeta_r)$ , and $\ell$ is a prime distinct from $r$ .

Specifically, the author seeks to establish conditions under which the image of the Galois representation $\rho_\ell: G_{\mathbb{Q}(\zeta_r)} \to \text{Aut}(J[\ell])$ is "large." In the context of abelian varieties with non-trivial endomorphism rings (specifically those with $\text{End}(J) \cong \mathbb{Z}[\zeta_r]$ ), "large" does not necessarily mean the full symplectic group $GSp_{2g}(\ell)$ , but rather the largest perfect subgroup of the centralizer of the automorphism $[\zeta_r]$ within the symplectic group.

Key Challenges:

Most existing results on large Galois images focus on abelian varieties over $\mathbb{Q}$ with trivial endomorphism rings or low dimensions (genus $\le 3$ ).
Previous methods often require the ground field to have no unramified extensions (class number 1) or require the curve to have everywhere semistable reduction.
This paper aims to handle curves over $\mathbb{Q}(\zeta_r)$ (which may have non-trivial class numbers for $r > 19$ ) and allows for bad reduction at specific primes, provided the coefficients are chosen carefully.

2. Methodology

The author employs a multi-faceted approach combining algebraic geometry, group theory, and arithmetic geometry:

A. $\lambda$ -adic Representations and Endomorphism Characters

The study is reduced from the mod $\ell$ representation to $\lambda$ -adic representations $\rho_\lambda$ for primes $\lambda | \ell$ in $\mathbb{Q}(\zeta_r)$ .
The determinant of these representations is linked to an algebraic Hecke character (the "endomorphism character"), denoted $\Omega$ . This character generalizes the CM character and is crucial for identifying the exact image of the Galois group, particularly the determinant part.

B. Group Theoretic Analysis

Centralizers: The image of the Galois group is constrained to the centralizer of the automorphism $[\zeta_r]$ in the symplectic group. The author utilizes results by Wall and others to describe these centralizers as products of General Linear groups ($GL$) or General Unitary groups ($GU$), depending on the inertia degree of $\ell$ in $\mathbb{Q}(\zeta_r)$ .
Maximal Subgroups: To prove the image is "large," the paper rules out containment in maximal subgroups of classical groups (Aschbacher classes $\mathcal{C}_1$ through $\mathcal{C}_8$ and the exceptional set $\mathcal{S}$ ). This involves analyzing elements with specific eigenvalue structures (e.g., transvections and elements with a single non-trivial eigenvalue of order $r$ ).
Generating Products: Using Goursat's Lemma and Ribet's Lemma, the author proves that if the projection of the image onto each factor of the product group is large, the total image is the full product.

C. Control of Inertia Groups (Newton Polytopes)

The paper utilizes the theory of $\Delta_v$ -regular curves (developed by Dokchitser) to control the action of inertia groups at primes of bad reduction.
By constructing polynomials $f(x)$ with specific Newton polytopes (defined by $v$ -degrees and heights), the author ensures that the inertia action at specific primes $p$ generates specific subgroups (e.g., containing transvections or elements with specific cycle structures).
This allows the author to construct curves with small coefficients (unlike previous methods using the Chinese Remainder Theorem which yielded huge coefficients) while maintaining control over the local Galois behavior.

D. Primitivity and Irreducibility

The paper establishes criteria for the representation to be irreducible and primitive.
A key innovation is handling fields with unramified extensions (when the class number of $\mathbb{Q}(\zeta_r)$ is odd but non-trivial). The author uses properties of the class number and root discriminants to rule out imprimitivity decompositions that would otherwise be possible.

3. Key Contributions

First Explicit Results for High-Dimensional Non- $GL_2$ Varieties: This is the first explicit result providing large Galois images for abelian varieties of dimension $g > 2$ that are not of $GL_2$ -type, specifically allowing the ground field to have unramified extensions (odd class number).
Exact Image Determination for $r=3$ : The paper provides the first accurate description of the full image of the mod $\ell$ representation for superelliptic curves with $r=3$ . It identifies the image as a specific extension of $GL_n(\ell)$ or $GU_n(\ell)$ by the cyclotomic character, determined by the endomorphism character.
Small Coefficient Examples: By using $\Delta_v$ -regularity, the author constructs explicit examples with relatively small coefficients, making the verification of large genus examples computationally feasible.
Generalization of the Endomorphism Character: The paper systematically studies the "endomorphism character" as a generalization of the CM character to determine the precise determinant of the Galois image.

4. Main Results

Theorem 1.2 (Main Theorem):
Let $r > 2$ be a prime with $\mathbb{Q}(\zeta_r)$ having an odd class number. Let $d$ be a degree divisible by $2r$ satisfying specific prime sum conditions. There exists a polynomial $f \in \mathbb{Q}(\zeta_r)[x]$ with coefficients satisfying congruence conditions such that for all primes $\ell$ outside a finite explicit set:

If the inertia degree $i$ of $\ell$ in $\mathbb{Q}(\zeta_r)$ is odd, the image contains $SL_n(\ell^i)^{(r-1)/2i}$ .
If $i$ is even, the image contains $SU_n(\ell^{i/2})^{(r-1)/i}$ .
Here $n = 2g/(r-1)$ .

Exact Image for $r=3$ (Theorem 7.5):
For $r=3$ and specific examples (e.g., $d=12, 18, 24, 30$ ):

If $\ell \equiv 1 \pmod 3$ : $\rho_\ell(G_{\mathbb{Q}(\zeta_3)}) = GL_n(\ell)^{\lceil g/3 \rceil, 6} \rtimes \langle \chi_\ell \rangle$ .
If $\ell \equiv 2 \pmod 3$ : $\rho_\ell(G_{\mathbb{Q}(\zeta_3)}) = GU_n(\ell)^{\lceil g/3 \rceil, 6} \rtimes \langle \chi_\ell \rangle$ .
The notation $GL_n(\ell)^{s,t}$ denotes matrices with determinants in a specific subgroup generated by powers of a generator and an element of order $t$ .

Explicit Examples (Section 7.1):
The paper constructs explicit curves with genera ranging from 10 to 36 (e.g., $y^3 = x^{12} + \pi x^{11} + \dots$ ) and lists the finite sets of "bad" primes where the image might not be large.

5. Significance and Applications

Inverse Galois Problem: The results provide a method to realize specific large classical groups (products of $SL_n$ and $SU_n$ ) as Galois groups over $\mathbb{Q}(\zeta_r)$ . This contributes significantly to the Inverse Galois Problem for these groups.
Advancement in Arithmetic Geometry: The work bridges the gap between the theory of abelian varieties with complex multiplication (or CM-type endomorphisms) and the study of large Galois images, which was previously dominated by the case of trivial endomorphism rings.
Computational Feasibility: By avoiding the "large coefficient" trap of previous CRT-based constructions, the paper opens the door to verifying these theoretical results on high-genus curves using standard computational tools.
Methodological Innovation: The integration of $\Delta_v$ -regularity with modern group-theoretic classifications of maximal subgroups offers a robust template for future studies on Galois representations of higher-dimensional varieties.

In summary, Pip Goodman's paper provides a rigorous framework and explicit constructions for superelliptic curves over cyclotomic fields that exhibit maximal possible Galois images, overcoming previous limitations regarding the base field's class number and the dimension of the abelian variety.