Goldfeld conjecture for non-hyperelliptic direction

1. Problem Statement and Motivation

The Goldfeld Conjecture Context:
The classical Goldfeld conjecture posits that for a given elliptic curve $E/\mathbb{Q}$, the analytic ranks of its quadratic twists $E(k)$ are distributed such that 50% have rank 0, 50% have rank 1, and 0% have rank $\ge 2$ (though infinitely many may exist). This "50-50-0" rule relies on the assumption that the twist family is governed by the hyperelliptic involution (for genus 1, all twists are quadratic).

The Gap:
For higher genus curves (specifically genus 2), a naive analogue of the Goldfeld conjecture fails.

1. Hyperelliptic Twists: Twists defined by $ky^2 = f(x)$ often exhibit rank biases (e.g., all having root number $+1$ or $-1$) due to the specific arithmetic of the hyperelliptic involution.
2. Non-Simple Jacobians: Many twist families of genus 2 curves have Jacobians that are isogenous to products of elliptic curves, reducing the problem to the elliptic curve case rather than a genuine genus 2 problem.
3. Lack of "Other Directions": For elliptic curves, the automorphism group is generically $\mathbb{Z}/2\mathbb{Z}$, meaning all twists are quadratic. To find a "non-hyperelliptic direction," one needs a curve with a larger automorphism group.

The Specific Problem:
The authors investigate the genus 2 curve $C_0: y^2 = x^6 + 1$, which has a large automorphism group ($2D_{12}$, order 24). They aim to:

1. Construct a twist family of $C_0$ that is not composed of hyperelliptic twists of each other.
2. Ensure the Jacobians of these twists are simple (not isogenous to products of elliptic curves).
3. Formulate and test an analogue of the Goldfeld conjecture for this specific "non-hyperelliptic direction."

2. Methodology

The paper employs a combination of arithmetic geometry, Galois theory, and analytic number theory.

A. Construction of the Twist Family

• Source Material: The authors utilize the classification of twists of $C_0$ by Cardona and Lario [CL20]. The twists are classified by conjugacy classes of subgroups of $\text{Aut}(\text{Aut}_{\mathbb{Q}}(C_0)) \cong C_2 \times D_{12}$.
• Selection: They focus on the conjugacy class $D_{12}^A$. Unlike the $C_2^A$ case (which yields hyperelliptic twists) or the $V_4^C$ case (which yields non-simple Jacobians), the $D_{12}^A$ case with a specific parameter choice ($k=1$) yields a family $\{C_d\}$ where:
  • No two distinct elements are hyperelliptic twists of each other.
  • The Jacobian $J_d$ is simple over $\mathbb{Q}$.
• Defining Equation: The family is defined by the equation:
  $$C_d: y^2 = 27\frac{d-3}{d+3} \left( x^6 - \frac{12d}{d-3}x^5 - 5dx^4 + \frac{40d^2}{3(d-3)}x^3 + \frac{5d^2}{3}x^2 - \frac{4d^3}{3(d-3)}x - \frac{d^3}{27} \right)$$
  for square-free integers $d$.

B. Arithmetic Invariants

To apply analytic methods, the authors compute necessary invariants:

• Splitting Fields: They determine the field $K_d$ where the automorphisms are defined and the field $L_d$ where the isomorphism to $C_0$ is defined. They prove $\text{Gal}(K_d/\mathbb{Q}) \cong S_3 \times C_2$.
• Simplicity of Jacobians: Using Honda-Tate theory and the Frobenius characteristic polynomial at specific primes, they prove $J_d$ is simple.
• Conductor Exponents: A crucial step involves computing the conductor of $C_d$ using the cluster picture method (Dokchitser et al.). They determine exact conductor exponents for primes $p | d$ and $p | (d+3)$, showing the reduction is tame for $p > 3$.

C. Analytic Approach (Explicit Formula)

The core of the proof relies on the explicit formula for $L$-functions (following Iwaniec–Kowalski).

• Goal: Estimate the average analytic rank $\frac{1}{|S(X)|} \sum_{d \in S(X)} g_d$.
• Technique: The average rank is bounded by the 1-level density of the zeros of the $L$-functions. This requires evaluating weighted sums of the traces of Frobenius, $a_p(C_d)$ and $a_{p^2}(C_d)$.
• Reduction to Elliptic Curves: Using results by Fité–Sutherland [FS14], the authors express $a_p(C_d)$ and $a_{p^2}(C_d)$ in terms of the trace of Frobenius of the elliptic curve $E_0: y^2 = x^3 + 1$ (which has CM by $\mathbb{Q}(\sqrt{-3})$).
• Cancellation: They analyze the distribution of the Frobenius traces based on the splitting behavior of the polynomial defining the twist. A key observation is that the contribution from $a_{p^2}$ vanishes for primes $p \equiv 1 \pmod 3$ due to symmetry, surviving only for $p \equiv 2 \pmod 3$.

3. Key Contributions and Results

Theorem 1.1 (Upper Bound on Average Rank)

Assuming the Generalized Riemann Hypothesis (GRH) for the $L$-functions of the family $\{C_d\}$, the authors prove an explicit upper bound for the average analytic rank.
Let $S(X)$ be the set of positive square-free integers $d < X$. Then:
$$\frac{1}{|S(X)|} \sum_{d \in S(X)} g_d \le \frac{1}{4} + \frac{6 + o(1)}{\sigma} + O\left( \frac{X^{\frac{3\sigma}{2} - \frac{1}{2}}}{\log X} + \frac{X^{\frac{\sigma}{2} - \frac{1}{2}}}{\log X} \right)$$
where $\sigma$ is the support of the Fourier transform of a test function. By optimizing $\sigma$, they obtain an upper bound of 18.25 (though the constant term is the focus).

Conjecture 1 (The Analogue of Goldfeld)

Based on the Katz–Sarnak philosophy (taking $\sigma \to \infty$), the authors conjecture that the true average analytic rank of this family is:
$$\lim_{X \to \infty} \frac{1}{|S(X)|} \sum_{d \in S(X)} g_d = \frac{1}{4}$$

Significance of the Constant 1/4

This is the most significant theoretical contribution.

• For elliptic curves, the average rank is expected to be $1/2$. This $1/2$ arises from a specific term in the explicit formula involving $a_{p^2}(E)$.
• For this genus 2 family, the term involving $a_{p^2}(C_d)$ vanishes for $p \equiv 1 \pmod 3$ due to the specific symmetries of the curve and the CM nature of the underlying elliptic curve $E_0$.
• Consequently, the contribution is halved, leading to the conjectured average of 1/4. This suggests that the "bias" in the root number distribution for non-hyperelliptic twists is fundamentally different from the hyperelliptic case.

Type $D_{12}^B$ (Negative Result)

The authors briefly analyze the $D_{12}^B$ twist family. They show that the Jacobians in this family are not simple (isogenous to a product of two elliptic curves). This confirms that the choice of $D_{12}^A$ was necessary to study a genuine genus 2 phenomenon.

4. Significance and Implications

1. New Direction in Arithmetic Statistics: The paper successfully identifies and analyzes a "non-hyperelliptic direction" in the space of twists. It demonstrates that the Goldfeld conjecture must be refined based on the specific geometric direction of the twist family, not just the genus.
2. Challenging the 1/2 Heuristic: The result provides a concrete counter-example to the intuition that the average rank of a "generic" family of abelian varieties should be $1/2$. It shows that symmetries in the automorphism group can force the average rank to be $1/4$.
3. Methodological Advancement: The paper integrates the cluster picture method for conductor computation with the explicit formula and CM theory in a sophisticated way, providing a template for analyzing other high-genus twist families.
4. Foundation for Future Work: While the paper proves an upper bound, the conjecture that the average is exactly $1/4$ remains open. This work sets the stage for proving the lower bound or finding a lower bound that matches the conjecture, which would be a major breakthrough in the understanding of $L$-functions for genus 2 curves.

In summary, Jeong and Park provide the first rigorous framework for studying the Goldfeld conjecture in a non-hyperelliptic setting, predicting a novel average rank of $1/4$ driven by the specific arithmetic symmetries of the curve $y^2 = x^6 + 1$.