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Geometric and arithmetic theta correspondences

Chao Li

TL;DR

The work surveys the trio of theta correspondences—classical, geometric, and arithmetic—within unitary groups, detailing how automorphic forms, cohomology of Shimura varieties, and algebraic cycles intertwine via the Weil representation, Siegel--Weil formulas, and doubling methods. It presents the arithmetic Kudla program, including modularity predictions for generating series of special cycles, arithmetic Siegel--Weil formulas, and Beilinson–Bloch-type height pairings that connect derivatives of L-functions to arithmetic heights. Recent progress on modularity results (e.g., for unitary orthogonal cases and certain imaginary quadratic fields) and the construction of integral models underpin many of these results, while open problems remain in higher codimension and full adelic arithmetic Siegel--Weil theories. Overall, the article articulates how the colorfully interconnected framework links central and derivative L-values to both geometric intersection theory and arithmetic heights, with wide-ranging implications for BSD-type conjectures and the arithmetic Gan–Gross–Prasad program.

Abstract

Geometric/arithmetic theta correspondences provide correspondences between automorphic forms and cohomology classes/algebraic cycles on Shimura varieties. We give an introduction focusing on the example of unitary groups and highlight recent advances in the arithmetic theory (also known as the Kudla program) and their applications. These are expanded lecture notes for the IHES 2022 Summer School on the Langlands Program.

Geometric and arithmetic theta correspondences

TL;DR

The work surveys the trio of theta correspondences—classical, geometric, and arithmetic—within unitary groups, detailing how automorphic forms, cohomology of Shimura varieties, and algebraic cycles intertwine via the Weil representation, Siegel--Weil formulas, and doubling methods. It presents the arithmetic Kudla program, including modularity predictions for generating series of special cycles, arithmetic Siegel--Weil formulas, and Beilinson–Bloch-type height pairings that connect derivatives of L-functions to arithmetic heights. Recent progress on modularity results (e.g., for unitary orthogonal cases and certain imaginary quadratic fields) and the construction of integral models underpin many of these results, while open problems remain in higher codimension and full adelic arithmetic Siegel--Weil theories. Overall, the article articulates how the colorfully interconnected framework links central and derivative L-values to both geometric intersection theory and arithmetic heights, with wide-ranging implications for BSD-type conjectures and the arithmetic Gan–Gross–Prasad program.

Abstract

Geometric/arithmetic theta correspondences provide correspondences between automorphic forms and cohomology classes/algebraic cycles on Shimura varieties. We give an introduction focusing on the example of unitary groups and highlight recent advances in the arithmetic theory (also known as the Kudla program) and their applications. These are expanded lecture notes for the IHES 2022 Summer School on the Langlands Program.
Paper Structure (22 sections, 9 theorems, 90 equations, 2 tables)

This paper contains 22 sections, 9 theorems, 90 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Classical theta correspondence (recap)
  3. Weil representation
  4. Theta lifting
  5. Siegel--Weil formula
  6. Rallis inner product formula
  7. Theta dichotomy in the equal rank case
  8. Geometric theta correspondence
  9. Unitary Shimura varieties
  10. Special cycles
  11. Kudla's generating function (arithmetic theta function)
  12. Geometric modularity
  13. Geometric theta lifting
  14. Geometric Siegel--Weil formula
  15. Geometric inner product formula
  16. ...and 7 more sections

Key Result

Theorem 2.3.1

Assume that the pair $(V,W)$ satisfies Weil's convergence condition. Then $E(g,s,\varphi)$ is holomorphic at $s_0$ and where $\kappa=1/2$ if $m>n$ and $\kappa=1$ otherwise.

Theorems & Definitions (26)

  • Theorem 2.3.1: Siegel--Weil formula
  • Theorem 2.4.1: Rallis inner product formula
  • Remark 3.1.1
  • Remark 3.1.2
  • Theorem 3.4.1: Geometric modularity
  • Remark 3.4.2
  • Theorem 3.6.1: Geometric Siegel--Weil formula
  • Theorem 3.7.1: Geometric inner product formula
  • Example 3.7.2
  • Conjecture 4.1.1: Arithmetic modularity
  • ...and 16 more