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A Tannakian description of the local Kaletha gerbe

Alexander Bertoloni Meli, Peter Dillery

TL;DR

The work constructs RigIsoc$_F$, a semisimple Tannakian category whose fiber-functor gerbe is canonically identified with Kaletha's Galois gerbe $\mathcal{E}_{Kal}$, thereby realizing the Kal gerbe in a Tannakian framework. The objects of RigIsoc$_F$ are explicit quadruples $(V,\tilde{\Phi},n,E)$ tied to finite Galois data and graded $L_E$-modules, assembled via four operations (products, modules, limits, descent) on isocrystal-like data, and organized so that their associated gerbe represents the class $-1$ in $\widehat{\mathbb{Z}} \cong H^{2}(\Gamma_{\overline{F}/F},u(\overline{F}))$. A Newton-map $f_X$ associates each simple object with a homomorphism from the Kaletha pro-torus $u$ into $\mathrm{GL}_{n}$, and basic objects correspond to elliptic twisted Levi centralizers $\mathrm{Res}_{E/F}(\mathrm{GL}_s)$; the simple-object classification is given by $\Gamma_{\tilde{E}/F}$-orbits of tuples in $(\mathbb{Q}/\mathbb{Z})^{\mathrm{Hom}_{F}(E',\overline{F})}$ satisfying slope-sum and coprimality conditions. The paper also relates RigIsoc$_F$ to Isoc$_F$ and Fargues' extended isocrystals, showing functorial links to the crystalline/Dieudonné framework and to a refined geometrization path for the local Langlands program. Altogether, this provides a concrete, computable description of the Kaletha Galois gerbe within a semisimple Tannakian setting and a complete parameterization of its simple objects via elliptic twisted Levi data.

Abstract

We construct, for a $p$-adic field $F$, an explicit semisimple Tannakian category $\text{RigIsoc}_{F}$ whose category of fiber functors recovers Kaletha's Galois gerbe $\mathcal{E}_{\text{Kal}}$. We then classify and write down the simple objects in $\text{RigIsoc}_{F}$, all of which come from elliptic twisted Levi subgroups of $\mathrm{GL}_{n}$.

A Tannakian description of the local Kaletha gerbe

TL;DR

The work constructs RigIsoc, a semisimple Tannakian category whose fiber-functor gerbe is canonically identified with Kaletha's Galois gerbe , thereby realizing the Kal gerbe in a Tannakian framework. The objects of RigIsoc are explicit quadruples tied to finite Galois data and graded -modules, assembled via four operations (products, modules, limits, descent) on isocrystal-like data, and organized so that their associated gerbe represents the class in . A Newton-map associates each simple object with a homomorphism from the Kaletha pro-torus into , and basic objects correspond to elliptic twisted Levi centralizers ; the simple-object classification is given by -orbits of tuples in satisfying slope-sum and coprimality conditions. The paper also relates RigIsoc to Isoc and Fargues' extended isocrystals, showing functorial links to the crystalline/Dieudonné framework and to a refined geometrization path for the local Langlands program. Altogether, this provides a concrete, computable description of the Kaletha Galois gerbe within a semisimple Tannakian setting and a complete parameterization of its simple objects via elliptic twisted Levi data.

Abstract

We construct, for a -adic field , an explicit semisimple Tannakian category whose category of fiber functors recovers Kaletha's Galois gerbe . We then classify and write down the simple objects in , all of which come from elliptic twisted Levi subgroups of .
Paper Structure (18 sections, 18 theorems, 90 equations)

This paper contains 18 sections, 18 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Motivation and overview
  3. Preliminary notation
  4. Acknowledgments
  5. Construction of Tannakian Categories
  6. Products of gerbes
  7. Limits of gerbes
  8. Modules
  9. Descent
  10. Concrete Tannakian categories
  11. Transition maps
  12. The category $\mathrm{RigIsoc}_{F}$
  13. The structure of $\mathrm{RigIsoc}_{F}$
  14. The Newton map
  15. Comparison with isocrystals
  16. ...and 3 more sections

Key Result

Theorem 1.1

There is a Tannakian category $\mathrm{RigIsoc}_{F}$ whose associated gerbe of fiber functors represents $[\mathcal{E}_{\mathop{\mathrm{Kal}}\nolimits}]$Of course, one can always take the Tannakian category $\mathop{\mathrm{Rep}}\nolimits(\mathcal{E}_{\mathop{\mathrm{Kal}}\nolimits})$ and recover $\ Morphisms betweem two such objects in $\mathrm{RigIsoc}_{F}$ are $W_{L_{E}/F}$-equivariant graded m

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Claim 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • Definition 2.6
  • ...and 29 more