Abstract
We provide an algorithm for computing special values modulo \(p^n\) of Dwork’s p-adic hypergeometric functions whose complexity increases at most \(O(n^4(\log n)^3)\). This is based on an explicit description of the Frobenius action on the rigid cohomology of a hypergeometric curve \((1-x^N)(1-y^M)=t\).
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Notes
- 1.
The referee pointed out to the author the recent article [Ke2], which almost solves the question completely (except the p-adic estimates on Frobenius). His approach is different from ours, more concise and maybe desirable.
- 2.
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Acknowledgements
The author would like to express sincere gratitude to Professors Alin Bostan and Kilian Raschel for encouraging the submission to this volume. He is also grateful to Professor Nobuki Takayama for the discussion on the bit complexity of the algorithm.
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Asakura, M. (2021). Frobenius Action on a Hypergeometric Curve and an Algorithm for Computing Values of Dwork’s p-adic Hypergeometric Functions. In: Bostan, A., Raschel, K. (eds) Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019. Springer Proceedings in Mathematics & Statistics, vol 373. Springer, Cham. https://doi.org/10.1007/978-3-030-84304-5_1
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