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.

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Mathematical Subject Classification 2010

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