Frobenius structure on rigid connections and arithmetic applications

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.