Frobenius structure on rigid connections and arithmetic applications
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914380952633344 |
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| author | Xu, Daxin Yi, Lingfei |
| author_facet | Xu, Daxin Yi, Lingfei |
| contents | 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 $θ$-connections arising from Vinberg's $θ$-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--Ngô--Yun. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_09252 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Frobenius structure on rigid connections and arithmetic applications Xu, Daxin Yi, Lingfei Number Theory Algebraic Geometry Representation Theory 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 $θ$-connections arising from Vinberg's $θ$-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--Ngô--Yun. |
| title | Frobenius structure on rigid connections and arithmetic applications |
| topic | Number Theory Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2603.09252 |