Counting odd genus $2$ curves with a marked rational $3$-torsion point

Technical Summary: Counting Odd Genus 2 Curves with a Marked Rational 3-Torsion Point

Problem Statement
This paper addresses the problem of counting genus 2 curves over the rational numbers $\mathbb{Q}$ that admit a monic Weierstrass model of odd degree and possess a Jacobian with a marked rational 3-torsion point. The curves are ordered by a "naive height" function. Specifically, the authors consider square-free, minimal polynomials $f(x) = x^5 + a_3x^3 + a_2x^2 + a_1x + a_0 \in \mathbb{Z}[x]$ defining the curve $C_f: y^2 = f(x)$. The height is defined as $H(f) = \max\{|a_3|^{1/4}, |a_2|^{1/6}, |a_1|^{1/8}, |a_0|^{1/10}\}$. The goal is to determine the asymptotic behavior of the number of pairs $(f, T)$, where $T$ is a non-zero rational 3-torsion point on the Jacobian $J_f$, as the height bound $X$ tends to infinity.

Methodology
The core of the methodology involves a parametrisation of the set of pairs $(f, T)$ using weighted homogeneous polynomials, reducing the counting problem to estimating the number of lattice points in a bounded region of a weighted projective space.

1. Parametrisation: The authors establish a bijection between pairs $(f, T)$ and tuples $P = (A, B, J, E)$ in a weighted projective space $\mathbb{P}(1, 2, 3, 4)$. This is achieved by constructing four weighted homogeneous polynomials $\Theta_0, \Theta_1, \Theta_2, \Theta_3 \in \mathbb{Z}[A, B, J, E]$ (with weights 4, 6, 8, and 10 respectively). These polynomials define the coefficients of the polynomial $f$ via the relation:
   $$729f(x) = 729x^5 + \Theta_3(P)x^3 + \Theta_2(P)x^2 + \Theta_1(P)x + \Theta_0(P).$$
   The marked 3-torsion point $T$ is represented by a divisor cut out by functions determined by $P$. This parametrisation is derived from the condition $G(x)^2 - A^2f(x) = H(x)^3$, where $G$ and $H$ are polynomials related to the divisor of the torsion point.

2. Geometry of Numbers: The problem is reduced to counting integer points $P = (A, B, J, E)$ within a compact region $R(X)$ defined by the height condition $H(F_P) \leq X$. The region $R(1)$ is shown to be compact because the polynomials $\Theta_i$ have no common zeros other than the origin.

3. Local Conditions and Sieving:

   • Integrality and Minimality: The authors derive necessary and sufficient congruence conditions on $P$ to ensure that the resulting polynomial $f$ has integer coefficients and is minimal (i.e., cannot be scaled down by a prime).
   • Primes $p \geq 5$: For primes $p \geq 5$, the condition for $f$ to be non-minimal is shown to be equivalent to $P$ lying in a specific sublattice where $p|A, p^2|B, p^3|J, p^4|E$. The density of minimal points at these primes is $1 - p^{-10}$.
   • Primes 2 and 3: The conditions for $p=2$ and $p=3$ are more complex and are handled via a finite set of congruence conditions, defining a local density constant $\beta_{2,3}$.

4. Asymptotic Counting: Using Davenport's Lipschitz principle (geometry of numbers), the authors count the lattice points in the scaled region $R(X)$. They apply a sieve to exclude points corresponding to non-minimal polynomials or degenerate cases (where the discriminant vanishes or $A=0$).

Key Contributions and Results
The main result is Theorem 1.1, which provides an exact asymptotic formula for the number of such curves:
$$\#W_{mark}(X) = cX^{10} + o(X^{10})$$
where $c$ is a positive, effectively computable constant given by:
$$c = \text{Vol}(R(1)) \cdot \beta_{2,3} \cdot \prod_{p \geq 5} (1 - p^{-10}).$$
Here, $\text{Vol}(R(1))$ is the volume of the compact region defined by the height function, and $\beta_{2,3}$ is the local density at primes 2 and 3.

The paper also provides Proposition 1.2, which explicitly constructs the weighted homogeneous polynomials $\Theta_i$ that parametrize the family of genus 2 curves with a marked 3-torsion point. This parametrisation is a significant technical achievement, as it compactifies the moduli space of such curves into a weighted projective space $\mathbb{P}(1, 2, 3, 4)$ in a way that extends the rational map to a morphism, allowing for the application of lattice point counting techniques.

Significance and Claims
The authors claim that this work is the first to determine an exact asymptotic formula for the number of curves with a prescribed level structure in a setting where the moduli variety is not a curve (specifically, a threefold in this case).

• Comparison to Previous Work: While counting elliptic curves with level structures (e.g., torsion points or isogenies) is well-established (citing works by Harron, Snowden, Boggess-Sankar, etc.), extending these results to higher genus curves is difficult. Previous work by Chan, Loughran, and Rome showed that almost all odd hyperelliptic curves have Jacobians with no prescribed level structure, but did not provide an asymptotic count for those that do.
• Methodological Innovation: The paper highlights the difficulty of compactifying the moduli space for higher-dimensional varieties. Unlike the elliptic curve case (where the modular curve is a curve and rational maps extend to morphisms automatically), the authors had to construct an "ad-hoc" compactification to $\mathbb{P}(1, 2, 3, 4)$ to ensure the map to the Weierstrass model was base-point-free.
• Scope: The result applies specifically to curves with a marked rational Weierstrass point (at infinity) and a marked rational 3-torsion point. The authors note in Remark 5.4 that counting curves with an unmarked 3-torsion point (forgetting the specific point $T$) would yield a count of $c/2 X^{10}$, as the rational 3-torsion subgroup is typically of order 3 (generated by $T$ and $-T$), except for a thin set of curves with a subgroup of order 9.

The paper does not propose experimental applications or future conjectures beyond the specific counting problem, maintaining a focus on the arithmetic geometry of the moduli space and the explicit construction of the parametrisation.