Thirteenth Algorithmic Number Theory Symposium ANTS-XIII University of Wisconsin, Madison July 16 – 20, 2018 Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII) July 16 – 20, 2018

Explicit Coleman integration in larger characteristic

Alex J. Best

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 [Har07] 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.

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