[Paper Review] Efficient computation of p-adic heights

David Harvey|ArXiv.org|Aug 24, 2007

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper presents a significant speedup in computing $p$-adic heights on elliptic curves over $\mathbf{Q}$, particularly for large $p$ and high $p$-adic precision. It improves the algorithm of Mazur, Stein, and Tate by optimizing the computation of the $p$-adic modular form $\mathbf{E}_2$ and the $p$-adic sigma function, reducing the running time to $\widetilde{O}(pN^2)$ for $\mathbf{E}_2$ mod $p^N$, and enabling feasible computation of thousands of $p$-adic digits even at $p \sim 10^{11}$.

ABSTRACT

We analyse and drastically improve the running time of the algorithm of Mazur, Stein and Tate for computing the canonical cyclotomic p-adic height of a point on an elliptic curve E/Q, where E has good ordinary reduction at p >= 5.

Motivation & Objective

• Address the lack of runtime bounds in the original Mazur–Stein–Tate algorithm for computing $p$-adic heights on elliptic curves.
• Enable practical computation of $p$-adic heights at high precision and for large primes $p \geq 5$ with good ordinary reduction.
• Provide tight bounds on required $p$-adic precision during intermediate computations to avoid unnecessary overhead.
• Optimize the asymptotic running time of the algorithm, especially for large $p$ and high-precision settings.
• Ensure the method is implementable on standard hardware, supporting applications in $p$-adic Birch and Swinnerton-Dyer conjecture research.

Proposed method

• Use Kedlaya’s algorithm to compute the matrix of Frobenius on Monsky–Washnitzer cohomology, enabling efficient computation of $\mathbf{E}_2(E,\omega)$ mod $p^N$.
• Apply a novel method to compute $\mathbf{E}_2(E,\omega)$ in $\widetilde{O}(pN^2)$ time, and $\widetilde{O}(p^{1/2}N^{5/2})$ time when $p > 6N$.
• Leverage $\mathbf{E}_2$ to compute the $p$-adic sigma function $\sigma_p(t)$ modulo the ideal $I_N = (p^r, p^{r-1}t, \dots, t^r)$ up to degree $N-1$.
• Compute $\sigma_p(t)$ mod $I_N$ in $\widetilde{O}(N^2 \log p)$ time, which is optimal up to logarithmic factors.
• Use the normalized $p$-adic height formula involving $\sigma_p(t)$, with a correction factor of $2p$ relative to the original normalization.
• Apply Theorem 3 to compute $h_p(P)$ mod $p^M$ in $\widetilde{O}(C_P + M\log^2 p + M^2\log p)$ time, given $\sigma_p(t)$ mod $I_{M'+1}$.

Experimental results

Research questions

• RQ1 What is the asymptotic running time of the Mazur–Stein–Tate algorithm for computing $p$-adic heights, and how can it be improved?
• RQ2 Can the computation of $\mathbf{E}_2(E,\omega)$ be accelerated for large primes $p$ and high $p$-adic precision?
• RQ3 How can the $p$-adic sigma function be computed efficiently modulo $I_N$, and what is the optimal time complexity?
• RQ4 What is the minimal $p$-adic precision required in intermediate steps to guarantee correct output at a given precision $M$?
• RQ5 Can the $p$-adic height of a rational point be computed in practice for $p \sim 10^{11}$ or for $M \sim 3000$ digits?

Key findings

• The computation of $\mathbf{E}_2(E,\omega)$ mod $p^N$ can be performed in $\widetilde{O}(pN^2)$ time, and in $\widetilde{O}(p^{1/2}N^{5/2})$ time when $p > 6N$, significantly improving upon prior methods.
• The $p$-adic sigma function $\sigma_p(t)$ can be computed modulo $I_N$ in $\widetilde{O}(N^2 \log p)$ time, which is optimal up to logarithmic factors.
• For $p = 10^{11} + 3$ and $M = 6$, the algorithm computes $h_p(P)$ in under 42 minutes for $\mathbf{E}_2$ and nearly instantaneously for the final height, avoiding the infeasible $10^{22}$-bit storage required by the original method.
• For $p = 5$ and $M = 3000$, the algorithm computes $\mathbf{E}_2$ in 229 seconds, $\sigma_p(t)$ in 158 seconds, and $h_p(P)$ in 6 seconds, demonstrating feasibility at high precision.
• The method avoids the $M^4$-cost power series expansions of the original algorithm, reducing the time for high-precision cases by a factor of up to $10^7$.
• The algorithm is implemented in SAGE and freely available, supporting practical use in $p$-adic $L$-function and regulator computations.