Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class

This paper constructs group-theoretic analogues of Johnson/Morita cocycles for pro-l groups to define Galois-cohomological classes for smooth curves, which are then used to demonstrate the existence of a non-hyperelliptic curve whose Ceresa class has a torsion image under the l-adic Abel-Jacobi map.

The Gist

Imagine you have a complex, multi-dimensional shape (a mathematical "curve") and you want to understand its hidden symmetries and structure. Mathematicians have long used a special tool called the Ceresa cycle to probe these shapes. Think of the Ceresa cycle as a "fingerprint" or a "DNA test" for the curve.

For a long time, mathematicians believed a specific rule: If a curve's fingerprint is "simple" (mathematically, "torsion"), the curve must be hyperelliptic.

What is a hyperelliptic curve? Think of it as a very symmetrical, "folded" shape, like a piece of paper folded perfectly in half. It's a special, easy-to-understand category of curves. The belief was that if your curve's fingerprint was simple, it had to be one of these folded, symmetrical shapes.

The Big Discovery:
This paper by Bisogno, Li, Litt, and Srinivasan proves that this rule is wrong. They found a curve that is not a simple, folded shape (it's "non-hyperelliptic"), yet its fingerprint is still "simple" (torsion). They broke the rule.

Here is how they did it, explained with analogies:

1. The New Tool: The "Group-Theoretic" Lens

Instead of looking at the curve directly (which is like trying to see a mountain through a thick fog), the authors built a new pair of glasses. They translated the geometry of the curve into the language of groups (mathematical structures that describe symmetry).

• The Old Way: Looking at the curve's physical shape and its relationship to its "Jacobian" (a complex machine that stores the curve's data).
• The New Way: They looked at the "fundamental group" of the curve. Imagine the curve is a tangled ball of string. The fundamental group is a code that describes all the possible ways you can loop a string around the knots in that ball.
• The Innovation: They created a new "Johnson class" (a specific code derived from these loops). This code is a more sensitive version of the old fingerprint. It can detect subtle features that the old tools missed.

2. The "Magic" Curve: The Fricke–Macbeath Curve

To prove their point, they needed a specific curve to test their new glasses on. They chose a famous, rare object called the Fricke–Macbeath curve.

• The Analogy: Imagine a 7-dimensional shape that is so perfectly symmetrical it has a "super-group" of symmetries (called $PSL_2(8)$) that acts on it. It's like a snowflake, but infinitely more complex and existing in higher dimensions.
• The Result: When they ran their new "Johnson class" test on this curve, the result was torsion (simple).
• The Twist: Despite having a "simple" fingerprint, this curve is not a folded, hyperelliptic shape. It is wild, complex, and non-hyperelliptic.

3. The "Genetic" Inheritance (The Genus 3 Curve)

The authors didn't stop there. They showed that this "simple fingerprint" trait is hereditary.

• The Analogy: Imagine a parent with a very specific, rare genetic trait. If you take a "child" curve that is a quotient (a simplified version) of this parent, the child inherits the trait.
• The Application: They took the complex 7-dimensional Fricke–Macbeath curve and "folded" it down using a specific symmetry (an order-2 element). This created a new curve of Genus 3 (a simpler shape, like a donut with three holes).
• The Surprise: This new, simpler curve is not hyperelliptic (it's not a simple fold), but it still has the "simple" torsion fingerprint.

Why Does This Matter?

For decades, mathematicians thought: "If the Ceresa cycle is trivial (simple), the curve must be hyperelliptic."

This paper says: "Nope."

They found the first known examples of curves that are not hyperelliptic but still have this "simple" property.

• For the experts: This confirms a prediction made by the Beilinson conjectures (a deep theory connecting geometry to numbers) and solves a question posed by Herbert Clemens.
• For the rest of us: It's like discovering a new species of animal that looks like a bird but has the DNA of a mammal. It forces us to rewrite the classification books and realize nature (or in this case, mathematics) is more diverse and surprising than we thought.

Summary

The authors built a new, sharper microscope (the Johnson class) to look at mathematical curves. They used it to find a "monster" curve (Fricke–Macbeath) and a "child" curve (Genus 3) that broke the old rules. They proved that you can have a complex, non-symmetrical curve that still possesses a "simple" mathematical fingerprint, shattering a long-held belief in the field.

Technical Summary

Here is a detailed technical summary of the paper "Group-theoretic Johnson classes and non-hyperelliptic curves with torsion Ceresa class" by Bisogno, Li, Litt, and Srinivasan.

1. Problem Statement and Context

The central object of study is the Ceresa cycle, a homologically trivial algebraic cycle $X - X^-$ in the Jacobian $Jac(X)$ of a smooth projective curve $X$ of genus $g \geq 3$. A classical result by Ceresa states that for a very general curve, this cycle is not algebraically trivial.

The paper investigates the Ceresa class, a Galois cohomology class $\mu(X, x)$ (and its basepoint-independent version $\nu(X)$) associated with this cycle via the $\ell$-adic cycle class map.

• Known Fact: If $X$ is hyperelliptic and $x$ is a Weierstrass point, the Ceresa class is trivial.
• Open Question: Does the triviality (or torsion nature) of the Ceresa class imply that the curve is hyperelliptic?
• Goal: The authors aim to construct group-theoretic analogues of these classes to analyze their properties, specifically seeking examples of non-hyperelliptic curves where the Ceresa class is torsion (i.e., has finite order in the relevant cohomology group).

2. Methodology

The authors develop a purely group-theoretic framework to define and analyze these classes, decoupling them from the specific geometry of the curve where possible.

A. Group-Theoretic Construction

Let $G$ be a finitely generated pro-$\ell$ group with torsion-free abelianization $G^{ab}$.

1. Augmentation Ideal: They consider the $\ell$-adic group ring $\mathbb{Z}_\ell[[G]]$ and its augmentation ideal $I$.
2. Modified Diagonal Class ($MD_{univ}$): They define a universal class in $H^1(\text{Aut}(G), \text{Hom}(I/I^2, I^2/I^3))$ associated with the extension of modules $0 \to I^2/I^3 \to I/I^3 \to I/I^2 \to 0$.
3. Johnson Class ($J_{univ}$): By analyzing the action of $G$ on this extension, they construct a quotient module $A(G)$ (the cokernel of a commutator map). They show that the restriction of the modified diagonal class to $G$ vanishes in $H^1(G, A(G))$, allowing the definition of a universal Johnson class $J_{univ} \in H^1(\text{Out}(G), A(G))$.
   • This construction refines existing work by Hain and Matsumoto (HM05), particularly for $\ell=2$, providing slightly more information.

B. Application to Curves

For a curve $X$ over a field $K$, they apply this construction to the pro-$\ell$ étale fundamental group $\pi_1^\ell(X_{\bar{K}})$.

• $MD(X, x)$: The pullback of $MD_{univ}$ via the Galois action on the fundamental group with a basepoint $x$.
• $J(X)$: The pullback of $J_{univ}$ via the outer Galois action (basepoint independent).
• Comparison: They prove that for smooth projective curves, these new classes correspond to the classes $\mu(X, x)$ and $\nu(X)$ defined by Hain and Matsumoto (specifically, $2MD(X,x) = \mu(X,x)$and$2J(X) = \nu(X)$ in the relevant cohomology).

C. Torsion Criteria

The authors establish criteria for these classes to be torsion:

1. Automorphism Invariance: If a finite subgroup $B \subset \text{Aut}(X)$ acts on the cohomology such that the invariant part of the coefficient module is zero ($H^0(B, A(G)) = 0$), then the Johnson class is torsion.
2. Dominance: They prove that if a curve $Y$ is dominated by a curve $X$ (i.e., there is a surjective map $X \to Y$), and $J(X)$ is torsion, then $J(Y)$ is also torsion. This generalizes the fact that a curve dominated by a hyperelliptic curve is hyperelliptic.

3. Key Contributions and Results

A. Theoretical Refinement

The paper provides a group-theoretic definition of the Johnson and Modified Diagonal classes that works for any pro-$\ell$ group with torsion-free abelianization, not just surface groups. This allows for a cleaner analysis of the $\ell=2$ case, where previous constructions were less precise.

B. Hyperelliptic Curves

They provide a concise group-theoretic proof that for any hyperelliptic curve, the Johnson class $J(X)$ is 2-torsion. This recovers the known fact that the Ceresa class is trivial for hyperelliptic curves.

C. Counterexamples to the "Torsion $\implies$ Hyperelliptic" Conjecture

The most significant result is the construction of non-hyperelliptic curves with torsion Ceresa classes, disproving the folk expectation that torsion Ceresa classes imply hyperellipticity.

1. The Fricke–Macbeath Curve:

   • Let $C$ be the genus 7 curve over a field of characteristic 0 with automorphism group $\text{PSL}_2(8)$.
   • Using character theory, the authors show that the action of $\text{PSL}_2(8)$ on the homology $H_1(C)$ has no non-trivial invariant maps to the relevant tensor products.
   • Result: $H^0(\text{PSL}_2(8), A(G)) = 0$, implying $J(C)$ is torsion. Since $C$ is not hyperelliptic (it has no central involution), this is a non-hyperelliptic example.

2. Genus 3 Quotients:

   • Using the "dominance" property, they take the quotient of the Fricke–Macbeath curve $C$ by an involution $\iota \in \text{PSL}_2(8)$.
   • Result: The quotient curve $C/\langle \iota \rangle$ is a non-hyperelliptic curve of genus 3 with a torsion Johnson class (and thus a torsion Ceresa class).

4. Significance and Implications

• Resolution of a Conjecture: The paper provides the first known explicit examples of non-hyperelliptic curves with torsion Ceresa classes, answering a question raised by Herbert Clemens and others regarding the relationship between the triviality of the Ceresa cycle and hyperellipticity.
• Connection to Beilinson Conjectures: As noted in Remark 1.3, Benedict Gross pointed out that the torsion nature of the Ceresa class for the Fricke–Macbeath curve is predicted by the Beilinson conjectures. The $L$-value controlling the rank of the relevant Chow group is non-zero, predicting a rank of zero (i.e., the cycle is torsion). This paper provides the geometric realization of this prediction.
• Methodological Advancement: The group-theoretic approach offers a robust tool for studying Galois actions on fundamental groups, independent of the specific model of the curve, and clarifies the relationship between different cohomological invariants (Hain-Matsumoto vs. Johnson).
• New Examples: The construction of the genus 3 counterexample is significant because genus 3 curves are the lowest genus where non-hyperelliptic curves exist, making this a minimal counterexample to the intuition that "torsion Ceresa class $\implies$ hyperelliptic."

In summary, the paper successfully bridges abstract group cohomology with arithmetic geometry to produce concrete counterexamples to a long-standing heuristic, demonstrating that the Ceresa class can be torsion for non-hyperelliptic curves, specifically those with large automorphism groups like the Fricke–Macbeath curve.