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 $\mathrm{\ell }$-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–Ngô–Yun.