Explicit Coleman Integration in Larger Characteristic

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.

16 pages, submitted to ANTS XIII, comments welcome!