p-adic hypergeometric differential equations and p-adic non-Liouville condition

2018/10/19 Fri 13:30 - 14:30
Room 152, Building No.3
Kazuaki Miyatani
Hiroshima University

The theme of this talk is the p-adic version of hypergeometric differential equations. In this talk, we prove that the arithmetic D-modules defined by the p-adic hypergeometric differential equations can be described as an iterated multiplicative convolution of arithmetic D-modules of rank one, under a "p-adic non-Liouvilleness" condition on the parameters. This is a p-adic counterpart of a result of N. M. Katz in the complex analytic case. Moreover, in the case where the parameters are rational, we use this result to prove that a finite field analogue of hypergeometric functions, which is introduced independently by Katz and Greene, naturally come from the arithmetic hypergeometric D-modules.