Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights

Technical Summary: Algorithms for Hyperelliptic Mumford Curves

Problem Statement
The paper addresses the scarcity of algorithmic tools for Mumford curves, which are $p$-adic analogues of Riemann surfaces that generalize Tate's uniformization of elliptic curves with split multiplicative reduction. While Tate uniformization is well-understood algorithmically, Mumford curves (genus $g \geq 2$ with split degenerate reduction) have lacked efficient computational methods. This gap hinders arithmetic applications such as computing $p$-adic Abelian integrals, $p$-adic heights, isogenies, and rational points via Chabauty-type methods. Specifically, existing algorithms for computing the underlying Schottky group from a curve equation were either restricted to "strong" geometric positions of fixed points (Kadziela's algorithm) or relied on naive, exponential-time approximations of theta functions.

Methodology
The authors develop a suite of algorithms for hyperelliptic Mumford curves defined over finite extensions of $\mathbb{Q}_p$ ($p \neq 2$). The methodology is divided into two main directions:

1. From Curve to Uniformization (Computing the Schottky Group):

   • Extension of Kadziela's Algorithm: The authors extend Kadziela's digit-by-digit approximation method for computing the fixed points of the Schottky group generators. They relax the geometric constraints on the fixed points from "strong Kadziela position" to "weak Kadziela position," allowing the algorithm to handle a broader class of curves (including those with specific stable reduction types).
   • Hensel Lifting: A novel method is introduced using multivariate Hensel lifting to compute the fixed points from the branch points of the hyperelliptic cover. This approach utilizes a coarse approximation of the Jacobian matrix (considering only words of length 0 and 1 in the theta function product) to achieve linear convergence. This method is significantly faster in practice than the digit-by-digit approach.
   • Theta Function Construction: The paper constructs explicit rigid analytic uniformizations using Whittaker groups (index 2 subgroups of discontinuous groups). It details the construction of theta functions $F$ and $H$ that map the $p$-adic upper half-plane to the curve, providing explicit formulas for the curve equation and the lifting of points.

2. From Uniformization to Arithmetic Applications:

   • Abelian Integration: The authors define an algorithm to compute $p$-adic Abelian integrals on the Jacobian of a Mumford curve. This relies on expressing the abelian logarithm in terms of $p$-adic theta functions. A key technical contribution is the construction of a "naive" $L$-invariant matrix derived from the period matrix $Q_W$ to correct for the lack of periodicity in the logarithm on the multiplicative torus $(K^\times)^g$.
   • $p$-adic Heights: The paper implements Werner's formula for computing local $p$-adic Schneider heights. Unlike previous methods requiring a model of the curve (which is difficult for high genus), this algorithm operates directly on the rigid uniformization. It involves computing an additive half-period matrix and evaluating theta pairings on divisors.

Key Contributions

• Generalized Schottky Group Computation: The paper provides the first practical algorithms to compute the Schottky group for hyperelliptic Mumford curves where the fixed points are in "weak Kadziela position," a condition satisfied by many curves that failed previous tests.
• Efficient Hensel Lifting: The introduction of a Hensel lifting technique for determining fixed points offers a linear convergence method that is computationally superior to previous exponential or digit-by-digit approaches.
• Model-Free Arithmetic: The algorithms for Abelian integrals and Schneider heights are formulated entirely in terms of the rigid analytic uniformization. This removes the dependency on finding a global algebraic model of the curve, a significant advantage for curves with bad reduction or high genus.
• Implementation: All algorithms are implemented in SageMath, leveraging existing packages for $p$-adic theta functions (Masdeu-Xarles) and local height computations (Magma).

Results
The authors validate their algorithms through several numerical examples:

• Verification of Previous Work: They successfully re-computed the Schottky groups and period matrices for six genus 2 curves from Kadziela's thesis, which were previously uncomputable by his original algorithm due to the fixed points not being in strong position.
• Consistency Checks: For a genus 2 curve over $\mathbb{Q}_5$, they verified that Abelian integrals vanish for torsion points (as predicted by theory) and confirmed their results against existing algorithms by Katz and Kaya.
• Schneider Heights: They computed local and global Schneider heights for genus 2 and genus 3 curves (including the modular curve $X_0(39)$). In the case of $X_0(39)$, they verified that the height pairing vanishes for a torsion divisor, demonstrating the correctness of the implementation.
• Regulator Computation: The paper successfully computed the Schneider regulator for a genus 2 modular abelian variety, confirming the quadraticity of the height pairing.

Significance and Claims
The paper claims to take a "major step toward making Mumford curves more algorithmically accessible." Its primary significance lies in bridging the gap between the theoretical theory of $p$-adic uniformization and practical arithmetic geometry. By providing model-free algorithms for Abelian integrals and $p$-adic heights, the authors enable the application of Chabauty–Coleman and Chabauty–Kim methods to Mumford curves without the need for explicit algebraic models. The authors explicitly state that these tools are particularly relevant for modular curves and Shimura curves, which often exhibit split degenerate reduction. They also note the potential for these algorithms to provide numerical evidence for higher-dimensional $p$-adic analogues of the Birch and Swinnerton-Dyer conjecture. The work is presented as a foundational step, with the authors noting that generalizations to superelliptic curves are expected but not yet fully developed in this text.