Geometrization of the local Langlands correspondence (2102.13459v4)

Abstract: Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the Fargues--Fontaine curve, prove a geometric Satake equivalence over the Fargues--Fontaine curve, and study the stack of $L$-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define $L$-parameters associated with irreducible smooth representations of $G(E)$, a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of $L$-parameters on the category of $\ell$-adic sheaves on $\mathrm{Bun}_G$.

Geometrization of the Local Langlands Correspondence

Introduction to the Local Langlands Correspondence

The Local Langlands Correspondence is a theoretical framework aiming to describe irreducible smooth representations $\pi$ of $G(E)$, where $G$ is a reductive group over a local field $E$. This correspondence is deeply rooted in number theory and geometry and has significant implications for understanding the symmetries and structure of various mathematical objects.

Geometrization via the Fargues--Fontaine Curve

Laurent Fargues and Peter Scholze propose a geometrization approach for the Local Langlands Correspondence through the Fargues--Fontaine curve. This curve provides a bridge between representations of $G(E)$ and $\ell$-adic sheaves on algebraic stacks, particularly focusing on the stack of $G$-bundles over the curve.

Main Contributions

1. Fargues--Fontaine Curve and $\ell$-Adic Sheaves

The paper develops foundational elements of the geometric Langlands program using the Fargues--Fontaine curve. It emphasizes the development of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the curve.

2. Geometric Satake Equivalence

A significant result established is the geometric Satake equivalence over the Fargues--Fontaine curve. This equivalence is crucial for translating problems in representation theory into problems in geometry, enabling the use of tools from algebraic geometry.

3. Stack of $L$-Parameters

The authors investigate the stack of $L$-parameters, providing insights into the spectral Bernstein center and its action on the category of $\ell$-adic sheaves.

Theoretical and Practical Implications

The Fargues-Scholze paper asserts the finiteness of cohomology for local Shimura varieties and moduli spaces of local shtukas. It suggests a map from the spectral Bernstein center to the Bernstein center and examines the spectral action of perfect complexes on the stack of $L$-parameters.

Theoretical Implications

The paper enhances the theoretical understanding of the connection between geometry and number theory through the Local Langlands Correspondence. It provides a robust framework for exploring $L$-packets and representation theory via geometric objects.

Practical Implications

This research opens new pathways for leveraging geometric methods in tackling problems in representation theory. It has potential applications in number theory, particularly in automorphic forms and algebraic geometry.

Future Directions

The geometric Langlands program proposed in this paper holds promise for future research in understanding the deeper ties between geometry and algebraic structures in number theory. Further exploration could lead to advancements in coherent sheaves and potentially new insights in related fields such as quantum computing and cryptography.

Conclusion

Fargues and Scholze's work presents a compelling case for the geometrization of the Local Langlands Correspondence. By bridging representation theory and algebraic geometry through the Fargues--Fontaine curve, they provide a novel approach to understanding $L$-parameters and $G(E)$-representations. This geometrization has the potential to unify disparate mathematical concepts, offering a fertile ground for future exploration in both pure and applied mathematics.