Frobenius structure on rigid connections and arithmetic applications

TL;DR

This paper constructs Frobenius structures on specific rigid irregular connections related to split simple groups, enabling the study of their monodromy, local systems, and verifying conjectures on rigidity and Langlands parameters.

Contribution

It introduces Frobenius structures on new classes of rigid irregular connections, linking p-adic and l-adic local systems, and verifies related conjectures on monodromy and rigidity.

Findings

Verified Reeder--Yu's prediction on epipelagic Langlands parameters.

Calculated the global geometric monodromy group of a special Airy local system.

Established cohomological and physical rigidity of the studied local systems.

Abstract

We construct the natural Frobenius structures on two families of rigid irregular $\check{G}$-connections on $\mathbb{G}_m$ (or $\mathbb{A}^1$) for a split simple group $\check{G}$: (i) the $\theta$-connections arising from Vinberg's $\theta$-groups introduced by Chen and Yun; (ii) the Airy connection of Jakob--Kamgarpour--Yi generalizing the classical Airy equations. These data form the $p$-adic companions of the $\ell$-adic local systems introduced by Yun and Jakob--Kamgarpour--Yi. Via the Frobenius structures, we study the local monodromy representations of these local systems at the unique wildly ramified point and verify the prediction of Reeder--Yu on epipelagic Langlands parameters in our setting. We calculate the global geometric monodromy group of a special Airy $\check{G}$-local system via its local monodromy. We show the cohomological rigidity and the physical rigidity of these local systems, as conjectured by Heinloth--Ng\^o--Yun.