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Courbes et fibrés vectoriels en théorie de Hodge $z$-adique globale

Siyan Daniel Li-Huerta

TL;DR

The paper constructs a global analogue of the Fargues--Fontaine framework for function fields by introducing the global Hartl--Pink curve and its moduli of $G$-bundles $\mathrm{Bun}_{G,F}$, linking them to the global Kottwitz set $B(F,G)$. It develops a comprehensive program: (i) vector bundles on the global curve via v-descent and a GAGA principle; (ii) a Tannakian description and semistable stratification of $G$-bundles with a localization map to local data; (iii) a fiber-product identity for global shtukas (Theorem D) that mirrors Scholze’s local theory and yields Langlands--Rapoport-type statements for shtuka moduli; (iv) geometric and sheaf-theoretic consequences, including a global conjecture F relating automorphic and spectral sides, and (v) a complete verification of Conjecture F in the commutative (torus) case, with a detailed account of Langlands duality via stacks of Galois representations and IndCoh on LS-stacks. A key technical achievement is a GAGA-type comparison over sousperfectoid spaces, enabling a robust bridge between algebraic and analytic worlds in this global setting.

Abstract

We study the global analogue of the Fargues-Fontaine curve over function fields $F$. We prove some foundational results about its moduli of $G$-bundles $\operatorname{Bun}_{G,F}$, which is a geometrization of the global Kottwitz set $B(F,G)$. For example, $\operatorname{Bun}_{G,F}$ plays the role of Igusa stacks over function fields. We use $\operatorname{Bun}_{G,F}$ to reformulate the global Langlands conjecture for $G$ over $F$ in terms of categorical local Langlands, refining conjectures of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky and Zhu. Finally, we verify this conjecture when $G$ is commutative. Along the way, we prove a GAGA theorem for smooth proper schemes over sousperfectoid spaces, which is of independent interest.

Courbes et fibrés vectoriels en théorie de Hodge $z$-adique globale

TL;DR

The paper constructs a global analogue of the Fargues--Fontaine framework for function fields by introducing the global Hartl--Pink curve and its moduli of -bundles , linking them to the global Kottwitz set . It develops a comprehensive program: (i) vector bundles on the global curve via v-descent and a GAGA principle; (ii) a Tannakian description and semistable stratification of -bundles with a localization map to local data; (iii) a fiber-product identity for global shtukas (Theorem D) that mirrors Scholze’s local theory and yields Langlands--Rapoport-type statements for shtuka moduli; (iv) geometric and sheaf-theoretic consequences, including a global conjecture F relating automorphic and spectral sides, and (v) a complete verification of Conjecture F in the commutative (torus) case, with a detailed account of Langlands duality via stacks of Galois representations and IndCoh on LS-stacks. A key technical achievement is a GAGA-type comparison over sousperfectoid spaces, enabling a robust bridge between algebraic and analytic worlds in this global setting.

Abstract

We study the global analogue of the Fargues-Fontaine curve over function fields . We prove some foundational results about its moduli of -bundles , which is a geometrization of the global Kottwitz set . For example, plays the role of Igusa stacks over function fields. We use to reformulate the global Langlands conjecture for over in terms of categorical local Langlands, refining conjectures of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky and Zhu. Finally, we verify this conjecture when is commutative. Along the way, we prove a GAGA theorem for smooth proper schemes over sousperfectoid spaces, which is of independent interest.
Paper Structure (9 sections, 63 theorems, 270 equations)

This paper contains 9 sections, 63 theorems, 270 equations.

Table of Contents

  1. Vector bundles on the global Hartl--Pink curve
  2. $G$-bundles
  3. Shtukas and the fiber product conjecture
  4. Geometric consequences
  5. Sheaf-theoretic consequences
  6. Relation with Langlands duality
  7. GAGA over sousperfectoid spaces
  8. $\ell$-adic realizations of Berkovich motives
  9. Global rigid inner forms, by Peter Dillery

Key Result

Lemma 1

For all adic spaces $\mathop{\mathrm{Spa}}\nolimits(A,A^+)$ over $S$, morphisms $\mathop{\mathrm{Spa}}\nolimits(A,A^+)\mathop{\rightarrow} Y^{\mathop{\mathrm{an}}\nolimits}_S$ over $S$ are equivalent to morphisms $\mathop{\mathrm{Spec}}\nolimits{A}\mathop{\rightarrow} Y$ over $D$.

Theorems & Definitions (151)

  • Definition
  • Remark
  • Definition
  • Lemma
  • proof
  • Lemma
  • proof
  • Proposition
  • proof
  • Proposition
  • ...and 141 more