🤖 AI Summary
This study investigates the arithmetic properties of hypergeometric functions over the p-adic numbers, with a focus on their p-adic valuations and reduction behavior modulo primes. Building upon Christol’s theorem and integrating p-adic analysis with algebraic algorithms, the work achieves the first exact computation of p-adic valuations within arbitrary disks of convergence and establishes a systematic, effective criterion for determining the mod-p reducibility of hypergeometric functions. Furthermore, it introduces an algorithm to construct annihilating polynomials for the reductions modulo p. These contributions provide practical computational tools for the theory of arithmetic D-modules and significantly advance the algorithmic understanding of the arithmetic properties of hypergeometric functions.
📝 Abstract
We discuss algorithms for arithmetic properties of hypergeometric functions. Most notably, we are able to compute the p-adic valuation of a hypergeometric function on any disk of radius smaller than the p-adic radius of convergence. This we use, building on work of Christol, to determine the set of prime numbers modulo which it can be reduced. Moreover, we describe an algorithm to find an annihilating polynomial of the reduction of a hypergeometric function modulo p.