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Arithmetic Hodge-Iwasawa Moduli Stacks

Xin Tong

TL;DR

Addresses the construction of arithmetic moduli stacks for $G$-bundles and $G$-isocrystals over arithmetic families of $FF_{K,toric}$ curves. The approach develops $(\infty,1)$-prestacks such as $\mathrm{Finiteproj}_{FF}$ and its filtrations, defines arithmetic Hecke stacks $H_{I,\dots}$ and shtuka moduli $\mathrm{Sht}_{I,\dots}$ via Frobenius twists and fiber products, and proves representability as Artin or derived stacks in both equal- and mixed-characteristic contexts. The contributions include explicit definitions of filtered and non-filtered variants, derived presentations, and parity with classical Hecke–shtuka constructions, plus extensions to variants of the Fargues-Fontaine setup. The work provides a coherent arithmetic moduli theory for Hodge-Iwasawa structures, with potential applications to Iwasawa deformation theory and $p$-adic geometric Langlands-type correspondences.

Abstract

In this article, we are going to construct arithmetic moduli stacks of $G$-bundles after our previous construction on Hodge-Iwasawa theory. These stacks parametrize certain Hodge-Iwasawa structures in a coherent way.

Arithmetic Hodge-Iwasawa Moduli Stacks

TL;DR

Addresses the construction of arithmetic moduli stacks for -bundles and -isocrystals over arithmetic families of curves. The approach develops -prestacks such as and its filtrations, defines arithmetic Hecke stacks and shtuka moduli via Frobenius twists and fiber products, and proves representability as Artin or derived stacks in both equal- and mixed-characteristic contexts. The contributions include explicit definitions of filtered and non-filtered variants, derived presentations, and parity with classical Hecke–shtuka constructions, plus extensions to variants of the Fargues-Fontaine setup. The work provides a coherent arithmetic moduli theory for Hodge-Iwasawa structures, with potential applications to Iwasawa deformation theory and -adic geometric Langlands-type correspondences.

Abstract

In this article, we are going to construct arithmetic moduli stacks of -bundles after our previous construction on Hodge-Iwasawa theory. These stacks parametrize certain Hodge-Iwasawa structures in a coherent way.
Paper Structure (5 sections, 4 theorems, 19 equations)

This paper contains 5 sections, 4 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Arithmetic Moduli Stacks
  3. Arithmetic Moduli Stacks of Filtered Frobenius Bundles
  4. Arithmetic Hecke Stacks, Arithmetic Moduli Stacks of Shtukas
  5. Variants

Key Result

Proposition 1

The prestack $\mathrm{Finiteproj}^{\mathrm{disc},\mathrm{Fil}}_{\mathrm{FF}^\mathrm{imper}_{K,\mathrm{toric}}}(-)$ fibered over $\mathrm{RigidAn}_{\mathbb{Q}_p,\mathrm{ffqc}}$ is an Artin stack. The parallel result in equal characteristic sitaution holds true as well.

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Definition 5
  • ...and 9 more