Irregular Hodge filtration of hypergeometric differential equations

TL;DR

This paper investigates the irregular Hodge filtrations of hypergeometric connections with rational parameters, providing a new geometric proof of known Hodge number calculations and linking hypergeometric sums to Frobenius Newton polygons.

Contribution

It offers a novel geometric approach to understanding irregular Hodge filtrations and confirms the equality of Frobenius Newton and irregular Hodge polygons for hypergeometric sums.

Findings

Hypergeometric sums are everywhere ordinary on the multiplicative group over finite fields.

A new geometric proof of the irregular Hodge numbers for hypergeometric connections.

Confirmation that Frobenius Newton polygon equals irregular Hodge polygon.

Abstract

Fedorov and Sabbah--Yu calculated the (irregular) Hodge numbers of hypergeometric connections. In this paper, we study the irregular Hodge filtrations on hypergeometric connections defined by rational parameters, and provide a new proof of the aforementioned results. Our approach is based on a geometric interpretation of hypergeometric connections, which enables us to show that certain hypergeometric sums are everywhere ordinary on $|\mathbb{G}_{m,\mathbb{F}_p}|$ (i.e. "Frobenius Newton polygon equals to irregular Hodge polygon").