Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arithmetic and Geometric Langlands Program

Xinwen Zhu

TL;DR

This survey explains how geometric Langlands ideas, notably the geometric Satake equivalence and global-local categorical correspondences, illuminate classical arithmetic Langlands and arithmetic geometry problems. It outlines a 2-categorical framework linking local and global Langlands parameters to categories of sheaves and automorphic data via the $L$-group ${}^L G$ (and the enhanced group ${}^cG$), excursion operators, and spectral actions. It highlights arithmetic applications to Shimura varieties, local models, congruence relations, generic Tate cycles, and the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives, including recent function-field results. The work synthesizes function-field proofs (e.g., Lafforgue) with conjectural, category-based formulations aimed at number fields, guided by geometric and physical insights.

Abstract

We explain how the geometric Langlands program inspires some recent new prospectives of classical arithmetic Langlands program and leads to the solutions of some problems in arithmetic geometry.

Arithmetic and Geometric Langlands Program

TL;DR

This survey explains how geometric Langlands ideas, notably the geometric Satake equivalence and global-local categorical correspondences, illuminate classical arithmetic Langlands and arithmetic geometry problems. It outlines a 2-categorical framework linking local and global Langlands parameters to categories of sheaves and automorphic data via the -group (and the enhanced group ), excursion operators, and spectral actions. It highlights arithmetic applications to Shimura varieties, local models, congruence relations, generic Tate cycles, and the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives, including recent function-field results. The work synthesizes function-field proofs (e.g., Lafforgue) with conjectural, category-based formulations aimed at number fields, guided by geometric and physical insights.

Abstract

We explain how the geometric Langlands program inspires some recent new prospectives of classical arithmetic Langlands program and leads to the solutions of some problems in arithmetic geometry.

Paper Structure

This paper contains 13 sections, 17 theorems, 63 equations.

Table of Contents

  1. From classical to geometric Langlands correspondence
  2. The geometric Satake
  3. Tamely ramified local geometric Langlands correspondence
  4. Global geometric Langlands correspondence
  5. From geometric to classical Langlands program
  6. Categorical arithmetic local Langlands
  7. Global arithmetic Langlands for function fields
  8. Geometric realization of Jacquet-Langlands transfer
  9. Applications to arithmetic geometry
  10. Local models of Shimura varieties
  11. The congruence relation
  12. Generic Tate cycles on mod $p$ fibers of Shimura varieties
  13. The Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

Key Result

Theorem 1.1.2

Let ${\operatorname{Gr}}_G:=LG/L^+G$ be the étale quotient of $LG$ by the (right) $L^+G$-action, which admits the left $L^+G$-action. Then ${\operatorname{Gr}}_G$ can be written as an inductive limit of $L^+G$-stable subfunctors $\underrightarrow\lim X_i$, with each $X_i$ being a perfect projective

Theorems & Definitions (37)

  • Remark 1.1.1
  • Theorem 1.1.2
  • Remark 1.1.3
  • Theorem 1.1.4
  • Remark 1.1.5
  • Remark 1.2.1
  • Theorem 1.2.2
  • Remark 1.2.3
  • Theorem 1.2.4
  • Remark 1.2.5
  • ...and 27 more