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High dimensional Gross--Zagier formula: a survey

Wei Zhang

Abstract

We survey recent developments on generalizing the Gross--Zagier formula to high dimensional Shimura varieties, with an emphasis on the Arithmetic Gan--Gross--Prasad conjecture and the relative trace formula approach.

High dimensional Gross--Zagier formula: a survey

Abstract

We survey recent developments on generalizing the Gross--Zagier formula to high dimensional Shimura varieties, with an emphasis on the Arithmetic Gan--Gross--Prasad conjecture and the relative trace formula approach.
Paper Structure (24 sections, 4 theorems, 105 equations)

This paper contains 24 sections, 4 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Gross--Zagier formula for Shimura curves
  3. AGGP conjecture for unitary Shimura varieties
  4. Special pairs of Shimura data and special cycles
  5. The main examples for this survey
  6. Gross--Zagier pair
  7. Gan--Gross--Prasad pair
  8. The arithmetic Gan--Gross--Prasad conjecture
  9. Height pairings
  10. The arithmetic Gan--Gross--Prasad conjecture
  11. The case of smooth integral models
  12. The case of regular integral models with parahoric levels
  13. The relative trace formula approach
  14. Hecke functions with regular supports
  15. $p$-adic height pairing and a $p$-adic AGGP conjecture
  16. ...and 9 more sections

Key Result

Theorem 3.14

Conjecture conj I=dJ holds for an inert $v$ with a hyperspecial level provided the residue characteristic is odd.

Theorems & Definitions (23)

  • Remark 2.1
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Conjecture 3.4
  • Conjecture 3.5
  • Conjecture 3.6
  • Conjecture 3.7
  • Remark 3.8
  • Remark 3.9
  • ...and 13 more