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The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group

This paper extends the Lang-Trotter conjecture to genus-2 curves with a reduced automorphism group containing $S_3$ , specifically establishing an average result for the family of curves $C_{\lambda}$ .

Original authors: Chihiro Ando, Shushi Harashita

Published 2026-04-02

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CC BY 4.0

Original authors: Chihiro Ando, Shushi Harashita

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 mystery that spans across the entire universe of numbers. The mystery involves curves (shapes drawn on a graph) and primes (the building blocks of numbers like 2, 3, 5, 7, etc.).

This paper is about a specific type of detective work involving Genus-2 curves. To understand this, let's break it down with some everyday analogies.

1. The Characters: Curves and Primes

Think of an Elliptic Curve (a simpler shape) as a smooth, looping rollercoaster track. Mathematicians love these tracks because they have hidden properties. One of the most interesting properties is whether the track becomes "supersingular" when you look at it through a specific lens (a prime number $p$ ).

Supersingular: Imagine the rollercoaster suddenly freezing or glitching in a very specific, rare way. It's a special state that doesn't happen often.
The Lang-Trotter Conjecture: In the 1970s, two mathematicians, Lang and Trotter, made a bold guess. They said: "If you look at a rollercoaster (an elliptic curve) through every prime number lens up to a certain size $X$ , the number of times it glitches (becomes supersingular) will be roughly proportional to $\frac{\sqrt{X}}{\log X}$ ."

Think of it like this: If you flip a coin $X$ times, you expect heads about half the time. But for these curves, the "glitches" are much rarer. The formula predicts exactly how rare they are.

2. The Problem: It's Too Hard to Solve Alone

The original Lang-Trotter conjecture for simple curves is still an unsolved mystery (like finding a needle in a haystack that keeps moving). However, mathematicians Fouvry and Murty found a clever workaround. Instead of looking at one specific rollercoaster, they looked at a whole family of them.

They asked: "If we pick a random rollercoaster from a huge bag of them, how many glitches do we see on average?"
They found that while we can't predict the glitches for one specific curve, the average behavior of the whole family follows a beautiful, predictable pattern.

3. The New Adventure: Moving to "Genus-2"

The authors of this paper, Chihiro Ando and Shushi Harashita, decided to level up the game. They moved from simple rollercoasters (Genus-1) to more complex, twisted shapes called Genus-2 curves.

The Analogy: If a Genus-1 curve is a simple loop (like a donut), a Genus-2 curve is like a figure-eight or a pretzel. It has more twists and turns.
The Specific Family: They focused on a very specific family of these pretzels, defined by a parameter $\lambda$ . You can think of $\lambda$ as a "knob" on a machine. Turning the knob changes the shape of the pretzel.
The Symmetry: These specific pretzels have a special symmetry group called $S_3$ (the symmetric group of 3 items). Imagine a triangle that can be rotated and flipped in 6 different ways and still look the same. These curves are special because they possess this specific kind of symmetry.

4. The Detective Work: Counting the Glitches

The authors wanted to know: On average, how many times does a pretzel from this family "glitch" (become superspecial) as we look at it through different prime lenses?

To solve this, they had to do some heavy lifting:

The Connection: They discovered that a Genus-2 pretzel glitches if and only if two specific, simpler rollercoasters (elliptic curves) hidden inside it also glitch. It's like saying a complex machine breaks only if two of its main gears break simultaneously.
The Counting: They counted how many times these hidden gears glitched for every prime number. This involved using ancient mathematical tools called Class Numbers (which count ways to arrange numbers in specific patterns) and Hilbert Class Polynomials (complex equations that act like maps to find these glitches).
The Result: They found a formula for the average number of glitches.

5. The Big Reveal (The Conclusion)

The paper concludes with a beautiful formula. Just like the original Lang-Trotter conjecture, the average number of glitches for these complex pretzels grows as the square root of the number of primes you check, divided by the log of that number.

However, the "constant" in front of the formula is different. It's a specific number involving $\pi$ and $\sqrt{3}$ (roughly $3.6$).

In plain English:

"We proved that if you take a huge collection of these specific, symmetrical, figure-eight-shaped curves and check them against millions of prime numbers, the average number of times they enter a 'superspecial' state follows a precise mathematical law. It's rare, but it happens often enough that we can predict the average perfectly."

Why Does This Matter?

Mathematical Beauty: It shows that even in the chaotic world of prime numbers and complex shapes, there is an underlying order and rhythm.
Cryptography: These curves are used in modern encryption. Understanding their "glitches" (superspecial points) helps cryptographers know which curves are safe to use and which might be vulnerable.
The "Average" Trick: It reinforces the idea that while individual mathematical objects might be unpredictable, large groups of them often behave in very regular, predictable ways.

Summary Metaphor:
Imagine a massive orchestra of instruments (the curves). You can't predict exactly when a single violin will squeak (glitch). But if you listen to the whole orchestra playing through a filter (the primes), you can predict exactly how many squeaks the whole group will make on average. This paper figured out the "squeak rate" for a new, more complex section of the orchestra.

1. Problem Statement

The paper addresses the extension of the Lang-Trotter conjecture from elliptic curves to genus-2 curves.

Background: For an elliptic curve $E/\mathbb{Q}$ without complex multiplication (CM), the Lang-Trotter conjecture posits that the number of primes $p < X$ where $E$ has supersingular reduction is asymptotically $c \sqrt{X}/\log X$ . While unproven for individual curves, Fouvry and Murty established an "average" version of this conjecture for families of elliptic curves.
Objective: The authors aim to establish a similar average estimation for a specific family of genus-2 curves. Specifically, they focus on curves $C_\lambda$ defined by:
$C_\lambda : y^2 = x(x-1)(x-\lambda)\left(x - \frac{\lambda-1}{\lambda}\right)\left(x - \frac{1}{1-\lambda}\right)$
These curves are characterized by having a reduced automorphism group containing the symmetric group $S_3$ . The goal is to compute the average number of primes $p < X$ for which $C_\lambda$ has a superspecial reduction (where the Jacobian is isomorphic to a product of supersingular elliptic curves) as the parameter $\lambda$ varies.

2. Methodology

The authors employ a combination of algebraic geometry, number theory, and analytic methods. The methodology proceeds in several stages:

A. Reduction to Elliptic Curves

Using a result by Ibukiyama, Katsura, and Oort, the condition for a genus-2 curve $C_\lambda$ to be superspecial is reduced to the condition that two associated elliptic curves, $E_{\Lambda_-}(\lambda)$ and $E_{\Lambda_+}(\lambda)$ , are both supersingular. These elliptic curves are defined over $\mathbb{F}_p$ (or $\mathbb{F}_{p^2}$ ) and depend on the parameter $\lambda$ .

B. Isogeny Construction and Endomorphism Rings

Isogenies: The authors explicitly construct degree-3 isogenies between $E_{\Lambda_-}(\lambda)$ and $E_{\Lambda_+}(\lambda)$ . They prove that the composition of these isogenies corresponds to the multiplication-by-$-3$ map.
Endomorphism Rings: By analyzing the Frobenius endomorphism and the constructed isogenies, they determine the structure of the endomorphism rings of these supersingular elliptic curves.
- For $p \equiv 1 \pmod 4$ , the endomorphism ring contains the order $\mathbb{Z}[\frac{1+\sqrt{-3p}}{2}]$ .
- For $p \equiv 3 \pmod 4$ (specifically $p \equiv 11 \pmod{12}$ ), the ring contains $\mathbb{Z}[\frac{1+\sqrt{-p}}{2}]$ .
- They prove that for $p \equiv 7 \pmod{12}$ , no such $\lambda$ exists (the count is zero).

C. Connection to Hilbert Class Polynomials

The authors link the $j$ -invariants of the supersingular curves $E_{\Lambda_\pm}(\lambda)$ to Hilbert class polynomials:

For $p \equiv 1 \pmod 4$ , the $j$ -invariants are roots of the Hilbert class polynomial $P_{3p}(X) \pmod p$ .
For $p \equiv 11 \pmod{12}$ , the $j$ -invariants are roots of $P_p(X) \pmod p$ .
A critical technical contribution is the factorization of $P_{3p}(X) \pmod p$ for $p \equiv 1 \pmod 4$ . The authors analyze the multiplicities of roots (specifically $8000$ and $54000$) and the structure of irreducible factors using modular polynomials $\Phi_n(X, Y)$ and the Gross-Zagier formula.

D. Counting via Graph Theory and Class Numbers

To count the number of $\lambda \in \mathbb{F}_p$ yielding superspecial curves ( $\psi_p$ ), the authors construct a graph $G_p$ where vertices are roots of the relevant Hilbert class polynomials and edges represent the relationship between $\lambda$ and the pair of $j$ -invariants $\{j(E_{\Lambda_-}), j(E_{\Lambda_+})\}$ .

By analyzing the weights of edges and the degrees of vertices in this graph, they derive exact formulas for $\psi_p$ in terms of class numbers $h(-3p)$ and $h(-p)$ .

E. Analytic Averaging

Finally, they apply the analytic techniques of Fouvry and Murty. They sum $\psi_p$ over primes $p < X$ weighted by the density of $\lambda$ in the range $|\lambda| \le N$ (or height $ht(\lambda) \le N$ ). This involves estimating sums of Dirichlet $L$ -functions $L(1, \chi)$ and applying the Polya-Vinogradov inequality and Abel's identity.

3. Key Contributions and Results

Theorem A: Exact Count over Finite Fields

The authors determine the exact number $\psi_p$ of parameters $\lambda \in \mathbb{F}_p$ for which $C_\lambda$ is superspecial:

If $p \equiv 1 \pmod 4$ : $\psi_p = \frac{3}{2} h(-3p)$ .
If $p \equiv 7 \pmod{12}$ : $\psi_p = 0$ .
If $p \equiv 11 \pmod{12}$ : $\psi_p = 3 h(-p)$ .
Significance: This provides the first explicit formula for the count of superspecial genus-2 curves of this specific $S_3$ -type in terms of class numbers, filling a gap left by previous works on other types (e.g., Klein-4 type).

Theorem B: Average for Integer Parameters

For $\lambda \in \mathbb{Z}$ with $|\lambda| \le N$ (where $N > X^{1+\epsilon}$ ), the average number of superspecial primes is:
$\frac{1}{N} \sum_{|\lambda| \le N} \phi_\lambda(X) \sim \frac{(6 + 4\sqrt{3})\pi}{9} \frac{\sqrt{X}}{\log X}$
This confirms the Lang-Trotter conjecture on average for this family, with a specific constant $c_{avg} = \frac{(6 + 4\sqrt{3})\pi}{9}$ .

Theorem C: Average for Rational Parameters

For $\lambda \in \mathbb{Q}$ with height $ht(\lambda) \le N$ , the average is:
$\frac{1}{N^2} \sum_{\substack{\lambda \in \mathbb{Q} \\ ht(\lambda) \le N}} \phi_\lambda(X) \sim \frac{4(3 + 2\sqrt{3})}{3\pi} \frac{\sqrt{X}}{\log X}$

4. Significance

Extension of Lang-Trotter: This is a significant step in generalizing the Lang-Trotter conjecture beyond elliptic curves to higher genus curves. It demonstrates that the "average" behavior holds for genus-2 curves with non-trivial automorphism groups.
New Arithmetic Invariants: The paper introduces a novel method of counting superspecial curves by relating them to the factorization of Hilbert class polynomials $P_{3p}(X)$ and constructing specific graphs of $j$ -invariants.
Factorization of Class Polynomials: The detailed analysis of the factorization of $P_{3p}(X) \pmod p$ for $p \equiv 1 \pmod 4$ is a standalone contribution to the theory of modular polynomials and complex multiplication.
Explicit Constants: The authors provide precise asymptotic constants involving $\sqrt{3}$ and $\pi$ , derived from the specific geometry of the $S_3$ -symmetric curves, distinguishing this family from the previously studied Klein-4 family.

In summary, the paper successfully extends the average Lang-Trotter conjecture to a specific family of genus-2 curves, utilizing deep connections between superspecial curves, isogenies, and class numbers to derive precise asymptotic formulas.