[Paper Review] Local-global compatibility over function fields

Siyan Daniel Li-Huerta|arXiv (Cornell University)|Jan 23, 2023

Advanced Algebra and Geometry4 citations

TL;DR

This paper establishes local-global compatibility for the global Langlands correspondence over function fields in positive characteristic, proving that V. Lafforgue's automorphic-to-Galois map commutes with Fargues–Scholze's semisimplified local Langlands correspondence. Using a uniformization morphism for moduli spaces of shtukas, the authors canonically lift the semisimplified correspondence to a non-semisimplified one and show that Fargues–Scholze's construction agrees with Genestier–Lafforgue's, resolving a longstanding question in the field.

ABSTRACT

We prove that V. Lafforgue's global Langlands correspondence is compatible with Fargues-Scholze's semisimplified local Langlands correspondence. As a consequence, we canonically lift Fargues-Scholze's construction to a non-semisimplified local Langlands correspondence for positive characteristic local fields. We also deduce that Fargues-Scholze's construction agrees with that of Genestier-Lafforgue, answering a question of Fargues-Scholze, Hansen, Harris, and Kaletha. The proof relies on a uniformization morphism for moduli spaces of shtukas.

Motivation & Objective

• To establish local-global compatibility between Lafforgue's global Langlands correspondence and Fargues–Scholze's semisimplified local Langlands correspondence over function fields in positive characteristic.
• To canonically lift Fargues–Scholze's semisimplified local Langlands correspondence to a non-semisimplified version for positive characteristic local fields.
• To resolve a question posed by Fargues–Scholze, Hansen, Harris, and Kaletha by proving that the Fargues–Scholze and Genestier–Lafforgue constructions agree in positive characteristic.
• To verify compatibility of the semisimplified local Langlands correspondence with the local Jacquet–Langlands correspondence in the positive characteristic setting.

Proposed method

• Utilizes a uniformization morphism for moduli spaces of global shtukas to relate local and global geometric structures.
• Relies on excursion operators and their compatibility across global and local settings via intersection cohomology of shtuka moduli stacks.
• Applies results from Xue on Weil group actions on compactly supported intersection cohomology of shtuka moduli spaces.
• Employs globalization techniques via Beuzart-Plessis, and ℓ-adic Kloosterman sheaves with their p-adic companions from Heinloth–Ngô–Yun and Xu–Zhu.
• Uses Deligne’s purity theorem to deduce irreducibility and purity of Galois representations, enabling comparison of L-parameters.
• Applies Bernstein center and Hecke eigensheaf techniques to compare automorphic and Galois side via Satake isomorphism and eigenvalue matching.

Experimental results

Research questions

• RQ1 Does Lafforgue’s global Langlands correspondence commute with Fargues–Scholze’s semisimplified local Langlands correspondence in positive characteristic function fields?
• RQ2 Can Fargues–Scholze’s semisimplified local Langlands correspondence be canonically lifted to a non-semisimplified version?
• RQ3 Do the Fargues–Scholze and Genestier–Lafforgue constructions of the local Langlands correspondence agree in positive characteristic?
• RQ4 Is the semisimplified local Langlands correspondence compatible with the local Jacquet–Langlands correspondence in positive characteristic?

Key findings

• Theorem A establishes that the global Langlands correspondence of Lafforgue is compatible with the semisimplified local correspondence of Fargues–Scholze, with the square of maps commuting.
• Theorem B shows that Fargues–Scholze’s semisimplified correspondence canonically lifts to a non-semisimplified local Langlands correspondence for positive characteristic local fields.
• Theorem C confirms that the Fargues–Scholze and Genestier–Lafforgue constructions of the local Langlands correspondence are identical in positive characteristic.
• Theorem D verifies that the semisimplified local Langlands correspondence commutes with the local Jacquet–Langlands correspondence for units of central simple algebras in positive characteristic.
• The proof of Theorem A is uniform across all reductive groups G, avoiding group-specific arguments used in prior works.
• The authors provide an abstract uniqueness argument showing that any local Langlands correspondence satisfying compatibility with parabolic induction, twisting by characters, and the local compatibility property must be unique, thereby confirming agreement between Fargues–Scholze and Genestier–Lafforgue.