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Frobenius structure on rigid connections and arithmetic applications

Daxin Xu, Lingfei Yi

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 $θ$-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.

Frobenius structure on rigid connections and arithmetic applications

Abstract

We construct the natural Frobenius structures on two families of rigid irregular -connections on (or ) for a split simple group : (i) the -connections arising from Vinberg's -groups introduced by Chen and Yun; (ii) the Airy connection of Jakob--Kamgarpour--Yi generalizing the classical Airy equations. These data form the -adic companions of the -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 -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.
Paper Structure (35 sections, 52 theorems, 174 equations)

This paper contains 35 sections, 52 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Rigid connections on curves: cohomological rigidity and physical rigidity
  3. Frobenius structures
  4. Application to monodromy representations and rigidity
  5. Organisation of the paper
  6. Review of $p$-adic differential equations
  7. Overconvergent ($F$-)isocrystals
  8. A theorem of Clark
  9. Rank one differential modules over the Robba ring, d'après Pulita Pulita
  10. $P$-adic isoclinic connections
  11. $P$-adic isoclinic connection over the open unit disc
  12. Comparison of local monodromy groups
  13. Frobenius structures on certain rigid connections
  14. Generalized Kloosterman sheaves, d'après Yun Yun16
  15. Comparison between $\mathop{\mathrm{Kl}}\nolimits_{\check{G}}^{\mathop{\mathrm{dR}}\nolimits}$ and $\mathop{\mathrm{Kl}}\nolimits_{\check{G}}^{\mathop{\mathrm{rig}}\nolimits}$
  16. ...and 20 more sections

Key Result

Theorem 1.2.3

Let $K=\mathbb{Q}_p(\mu_p)$, $\overline{K}$ an algebraic closure of $K$ and set $\lambda=-\pi\in K$ satisfying $\pi^{p-1}=-p$. Assume $(p,m)=1$ (resp. $p>h$) and that the Kac coordinate $s_0=1$ and the same holds for the order $m$ inner stable grading of the Lie algebra $\mathfrak{g}$ of the dual gr

Theorems & Definitions (107)

  • Theorem 1.2.3: \ref{['ss:Frob-theta-connection']},\ref{['ss:compare-dR-rig-Airy']}
  • Remark 1
  • Theorem 1.3.2: \ref{['c:Swan-conductor']}, \ref{['p:Airy-coh-rig']}
  • Corollary 1
  • Remark 2
  • Theorem 1.3.4: \ref{['t:monodromy-Ai']}
  • Remark 3
  • Theorem 1.3.6: \ref{['c:strong-phy-rigidity']}
  • Theorem 1.3.8: \ref{['t:physical-rig-l']}
  • Theorem 2.2.1: Clark Theorem 3, Set97
  • ...and 97 more