Computing p-adic heights on hyperelliptic curves

Technical Summary: Computing $p$-adic Heights on Hyperelliptic Curves

Problem Statement
The paper addresses the computational challenge of determining the local Coleman–Gross $p$-adic height pairing $h_p$ on the Jacobian $J$ of a hyperelliptic curve $X: y^2 = f(x)$ over $\mathbb{Q}$, where $p$ is a prime of good reduction. While global $p$-adic heights are sums of local contributions, the local heights at primes $v \neq p$ are computable via intersection theory on arithmetic surfaces. However, the local height $h_p$ at $p$ requires $p$-adic analysis (Coleman integration).

Previously, the only available algorithm for computing $h_p$ was that of Balakrishnan and Besser [BB12]. This method was restricted to curves possessing an odd-degree model over $\mathbb{Q}_p$. Consequently, it could not be directly applied to hyperelliptic curves with even-degree models, limiting the scope of applications such as the quadratic Chabauty method for integral points and numerical verification of $p$-adic Birch and Swinnerton–Dyer (BSD) conjectures for such curves.

Methodology
The authors propose a new algorithm to compute $h_p$ that works for both odd and even-degree models, though it is specifically optimized for the even-degree case. The core strategy involves reducing the computation of $h_p(P-Q, R-S)$ to the evaluation of Coleman integrals of specific differential forms.

1. Reduction to Even Degree: The algorithm assumes the curve is given by an even-degree model $y^2 = f(x)$ with $f$ monic. This ensures the existence of two distinct points at infinity, $\infty^+$ and $\infty^-$, and a non-trivial degree-zero divisor supported at infinity, $D_\infty = \infty^- - \infty^+$.
2. Decomposition of Divisors: Using the bi-additivity of the height pairing, the problem reduces to computing $h_p$ for pairs of points. The authors distinguish two cases:
   • Case 1: One divisor is $D_\infty$. The computation of $h_p(\infty^- - \infty^+, R-S)$ is reduced to Coleman integrals of the differential $\omega_g = \frac{x^g dx}{y}$. The authors utilize the action of Frobenius on Monsky–Washnitzer cohomology to determine the necessary coefficients to project the differential associated with $D_\infty$ onto the chosen isotropic subspace $W$.
   • Case 2: Affine Points. For $h_p(P-Q, R-S)$ with affine points, the authors employ a change of variables $\tau: C \to C'$ that maps the divisor $P - \iota(P)$ (where $\iota$ is the hyperelliptic involution) to $D_\infty$ on a transformed curve $C'$. This transformation effectively converts the affine case into the "infinite" case (Case 1), allowing the reuse of the efficient computation for $D_\infty$.
3. Implementation Details: The algorithm relies on precomputed data: a basis for $H^1_{dR}(C/\mathbb{Q}_p)$, the Frobenius action matrix, the cup product matrix, and a basis for an isotropic complementary subspace $W$. The actual height computation involves linear algebra to find coefficients and the evaluation of Coleman integrals using Balakrishnan's existing implementation for even-degree curves.

Key Contributions

• Generalization to Even Degree: The primary contribution is an algorithm that computes $p$-adic heights for hyperelliptic curves with even-degree models, a scenario previously inaccessible to the standard Balakrishnan–Besser algorithm.
• Efficiency and Simplicity: The proposed method is significantly faster and simpler than the [BB12] approach. It avoids the high-precision computations in local coordinates over high-degree extensions of $\mathbb{Q}_p$ required by the previous method for odd-degree curves, instead leveraging the structure of the even-degree model and the divisor at infinity.
• Precision Analysis: The paper provides a rigorous precision analysis, establishing the necessary working precision to achieve a target accuracy, accounting for the determinants of the Frobenius and cup product matrices.
• SageMath Implementation: A complete implementation is provided in SageMath, utilizing existing libraries for Monsky–Washnitzer cohomology and Coleman integration.

Results
The authors present timing comparisons demonstrating the superiority of their algorithm:

• For a genus 2 curve at $p=7$ with 10-digit precision, the new algorithm takes 2 seconds compared to 7 seconds for [BB12].
• For higher precision (300 digits) and larger primes (e.g., $p=503$), the [BB12] implementation fails to complete within a week, whereas the new algorithm succeeds (e.g., 4 minutes for $p=503$).
• The algorithm successfully handles curves of higher genus (up to 17 in the examples) and high precision, which were previously intractable.

Significance and Applications
The paper highlights three specific applications enabled by this new algorithm:

1. Quadratic Chabauty for Rational Points: The method accelerates the computation of rational points on curves where the Mordell–Weil rank equals the genus. It allows for the use of smaller primes (e.g., $p=7$ for $X_0^+(107)$) that were previously unusable due to the odd-degree model restriction, drastically reducing computation time (from 40 minutes to 47 seconds in a specific example).
2. Quadratic Chabauty for Integral Points: It facilitates a new, simpler method for computing integral points on even-degree hyperelliptic curves over number fields. This approach avoids the technical complexities of $p$-adic Arakelov theory and double integrals required by previous methods, making it applicable to curves that were previously out of reach.
3. Numerical Evidence for $p$-adic BSD: The algorithm enables the numerical verification of the $p$-adic Birch and Swinnerton–Dyer conjecture for modular abelian varieties of $GL_2$-type where the associated curve lacks an odd-degree model over $\mathbb{Q}_p$. The authors verify the conjecture for the Jacobian of $X_0^+(67)$ at $p=11$, a case where the previous algorithm could not be applied.

The work is modest in its claims, noting that while the algorithm assumes points are defined over $\mathbb{Q}_p$ (due to implementation limitations of Coleman integration over extensions), the theoretical framework allows for potential future extensions. The authors emphasize that their work provides the necessary computational tools to expand the scope of explicit Diophantine geometry and $p$-adic conjecture testing.