An algorithm of computing special values of Dwork's p-adic hypergeometric functions in polynomial time

2019/09/06 by Masanori Asakura, Asakura, Masanori
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1909.02700

openalex publication_date 2019/09/06 · openalex created_date 2019/09/12 · openalex updated_date 2026/07/28

Abstract

Dwork's p-adic hypergeometric function is defined to be a ratio _sF_s-1(t)/_sF_s-1(t^p) of hypergeometric power series. Dwork showed that it is a uniform limit of rational functions, and hence one can define special values on |t|_p=1. However to compute the value modulo pⁿ in the naive method, the bit complexity increases by exponential when n→∞. In this paper we present a certain algorithm whose complexity increases at most O(n⁴(log n)³).

Citations

• F-isocrystal and syntomic regulators via hypergeometric functions
• New p-adic hypergeometric functions and syntomic regulators

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