Linear quadratic Chabauty

Technical Summary: Linear Quadratic Chabauty

Problem Statement
The paper addresses the computation of integral points $U(\mathbb{Z})$ (or $U(\mathcal{O}_K)$ over number fields) on hyperelliptic curves $X: y^2 = f(x)$. Specifically, it targets curves of even degree $2g+2$ where the leading coefficient of $f$ is a perfect square. The authors operate under the standard Chabauty condition where the Mordell–Weil rank $r$ of the Jacobian $J$ equals the genus $g$ (or satisfies a generalized bound over number fields).

Existing methods for integral points, such as the quadratic Chabauty method developed by Balakrishnan, Besser, and Müller [BBM16, BBM17], rely on constructing a non-trivial locally analytic function $\rho$ whose values on integral points lie in a finite, computable set. However, these existing approaches are technically complex, requiring tools such as $p$-adic Arakelov theory, extensions of local heights to divisors with common support, double Coleman integrals, and integrals with tangential base points. Furthermore, the computation of double Coleman integrals is identified as the primary bottleneck in terms of speed.

Methodology
The authors introduce a "Linear Quadratic Chabauty" method that simplifies the construction of the function $\rho$ by exploiting the specific geometry of even-degree curves with square leading coefficients.

1. Geometric Setup: Since the leading coefficient of $f$ is a square, there exists a non-trivial $\mathbb{Q}$-rational divisor of degree 0 supported at the two points at infinity, denoted $\infty^-$ and $\infty^+$.
2. Linearization of the Height: The method utilizes the global $p$-adic height pairing $h$ (Coleman–Gross). The authors observe that the linear functional $\lambda(a) = h(\infty^- - \infty^+, a)$ on the Jacobian can be expressed as a linear combination of Coleman integrals of holomorphic differentials $\omega_i = x^i dx/y$. This is possible because the space of linear maps $J(\mathbb{Q}) \otimes \mathbb{Q}_p \to \mathbb{Q}_p$ is spanned by these integrals when $r=g$.
3. Construction of $\rho$: Instead of using double integrals, the authors define the function $\rho: U(\mathbb{Z}_p) \to \mathbb{Q}_p$ as:
   $$\rho(P) = \sum_{i=0}^{g-1} \alpha_i \int_Q^P \omega_i - h_p(\infty^- - \infty^+, P - Q)$$
   where $Q$ is a known integral point, $\alpha_i$ are coefficients determined by the linear relation of the height, and $h_p$ is the local height at $p$.
4. Restriction to Integral Points: For an integral point $P$, the global height relation implies:
   $$\rho(P) = \sum_{q \neq p} h_q(\infty^- - \infty^+, P - Q)$$
   The authors analyze the local heights $h_q$ for $q \neq p$. They prove that for good primes, these heights vanish, and for bad primes, they take values in a finite, computable set $T_q$ derived from intersection theory on a regular model of the curve. Thus, $\rho(P)$ is constrained to a finite set $T = \sum T_q$.
5. Generalization to Number Fields: The method is extended to number fields $K$ of degree $d$. By using multiple independent continuous idèle class characters $\chi$, the authors generate $d - r_K$ additional equations (where $r_K$ is the rank of the unit group). Combined with $r_K$ equations from restriction of scalars, this yields a system of $d$ equations sufficient to cut out the integral points under the condition $\text{rk}(J/K) \leq dg - r_K$.

Key Contributions

• Simplification: The method eliminates the need for double Coleman integrals, tangential base points, and $p$-adic Arakelov theory. It relies solely on single Coleman integrals and standard local height computations.
• Efficiency: By avoiding double integrals, the method is significantly faster than previous quadratic Chabauty implementations.
• New Applicability: It provides a practical algorithm for even-degree hyperelliptic curves with square leading coefficients, a case where previous methods were either inapplicable or required more complex machinery.
• Number Field Extension: The paper generalizes the technique to compute $\mathcal{O}_K$-integral points, utilizing a combination of linear relations from restriction of scalars and $p$-adic heights associated with idèle class characters.

Results and Examples
The authors provide explicit implementations and examples:

• Over $\mathbb{Q}$: They compute the integral points on a genus 2 curve $y^2 = x^6 + 2x^5 - 7x^4 - 18x^3 + 2x^2 + 20x + 9$. The method successfully isolates the integral points, proving the set is complete.
• Over $\mathbb{Q}(\sqrt{7})$: They determine the $\mathcal{O}_K$-integral points on a genus 2 curve defined over the real quadratic field $K = \mathbb{Q}(\sqrt{7})$. This example involves a rank 3 Jacobian and requires the combination of the new linear quadratic Chabauty method with the Mordell–Weil sieve. The authors note that no other known method could handle this specific example.

Significance and Claims
The paper claims that this approach is "significantly simpler and faster" than all existing versions of the quadratic Chabauty method. The primary significance lies in the reduction of technical overhead: by restricting the $p$-adic height to a linear function via the divisor at infinity, the authors bypass the most computationally expensive and theoretically intricate components of the previous framework.

The authors are modest regarding the scope, acknowledging that:

• The method requires the leading coefficient of $f$ to be a square (or a negative square).
• For non-square leading coefficients, the extension of the method would still require the more complex tools (double integrals, Arakelov theory) that this paper aims to avoid.
• The resulting set of points may not always be finite or equal to the set of integral points without the aid of the Mordell–Weil sieve.
• The method is currently limited to primes that split completely in the number field case, though the theoretical results hold more generally.

The work is presented as a practical improvement for a specific but important class of curves, offering a more accessible and efficient tool for computing integral points in the context of the Chabauty–Coleman method.