Irregular Hodge filtration of hypergeometric differential equations

Qin, Yichen; Xu, Daxin

doi:10.2140/ant.2025.19.2481

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Irregular Hodge filtration of hypergeometric differential equations

Yichen Qin and Daxin Xu

Vol. 19 (2025), No. 12, 2481–2514

DOI: 10.2140/ant.2025.19.2481

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 $| 𝔾_{m, 𝔽_{p}} |$ ; i.e., “Frobenius Newton polygon equals the irregular Hodge polygon”.

Keywords

hypergeometric connections, irregular Hodge filtration, p-adic slopes, exponential sums

Mathematical Subject Classification

Primary: 14D07

Secondary: 11T23, 14F30, 14F40, 33C15

Milestones

Received: 10 May 2024

Revised: 14 October 2024

Accepted: 5 December 2024

Published: 20 October 2025

Authors

Yichen Qin
School of Mathematical Sciences Fudan University Shanghai China

Daxin Xu
Morningside Center of Mathematics and State Key Laboratory of Mathematical Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

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Vol. 19, No. 12, 2025