#### Vol. 2, 2019

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Explicit Coleman integration in larger characteristic

### Alex J. Best

Vol. 2 (2019), No. 1, 85–102
##### 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 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.

##### Keywords
Coleman integration, hyperelliptic curves, Kedlaya's algorithm
##### Mathematical Subject Classification 2010
Primary: 11G20
Secondary: 11Y16, 14F30
##### Milestones
Revised: 8 June 2018
Accepted: 9 September 2018
Published: 13 February 2019
##### Authors
 Alex J. Best Department of Mathematics and Statistics Boston University Boston, MA United States