Explicit Coleman Integration in Larger Characteristic

TL;DR

This paper introduces a more efficient algorithm for computing p-adic Coleman integrals on hyperelliptic curves in large characteristic, utilizing fast recurrence techniques to improve performance.

Contribution

The paper presents a novel, faster algorithm for p-adic Coleman integrals that leverages linear recurrence methods in Monsky-Washnitzer cohomology, reducing computational complexity.

Findings

Algorithm has quasilinear complexity in √p

Implementation outperforms existing methods

Effective for large primes p and high genus

Abstract

We describe a more efficient algorithm to compute p-adic Coleman integrals on odd degree hyperelliptic curves for large primes p. The improvements come from using fast linear recurrence techniques when reducing differentials in Monsky-Washnitzer cohomology, a technique introduced by Harvey arXiv:math/0610973 when computing zeta functions. The complexity of our algorithm is quasilinear in $\sqrt p$ and is polynomial in the genus and precision. We provide timings comparing our implementation with existing approaches.