Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang

Gao, Ziyang; Känel, Rafael von; Mocz, Lucia

doi:10.23943/princeton/9780691193779.003.0004

Arithmetic and Geometry: Ten Years in Alpbach (AMS-202)

Gisbert Wüstholz (ed.), Clemens Fuchs (ed.)

https://doi.org/10.23943/princeton/9780691193779.001.0001

Published online:

21 May 2020

Published in print:

08 October 2019

Online ISBN:

9780691197548

Print ISBN:

9780691193779

- Preface
  - Notes
  - Notes
One Introduction
- Notes
- Notes
Two Local Shimura Varieties: Minicourse Given by Peter Scholze
- Notes
- Notes
Three Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier
- Notes
- Notes
- 4.1 Heights and L-Functions 4.1 Heights and L-Functions
  - 4.1.1 The Faltings Height and the Generalized Szpiro Conjecture 4.1.1 The Faltings Height and the Generalized Szpiro Conjecture
    - The Faltings height. The Faltings height.
    - Generalized Szpiro conjecture. Generalized Szpiro conjecture.
  - 4.1.2 CM Abelian Varieties and their Faltings Height 4.1.2 CM Abelian Varieties and their Faltings Height
    - Theory of complex multiplication. Theory of complex multiplication.
    - The generalized Szpiro conjecture for CM abelian varieties. The generalized Szpiro conjecture for CM abelian varieties.
    - The Colmez conjecture. The Colmez conjecture.
    - Averaged Colmez conjecture. Averaged Colmez conjecture.
    - Applications of the Colmez conjectures. Applications of the Colmez conjectures.
  - 4.1.3 The Function Field Setting: The Work of Yun-Zhang 4.1.3 The Function Field Setting: The Work of Yun-Zhang
    - A brief introduction to shtuka. A brief introduction to shtuka.
    - Heegner-Drinfeld cycles and the intersection problem. Heegner-Drinfeld cycles and the intersection problem.
    - Significance and open questions. Significance and open questions.
- 4.2 Shimura Curves and Averaged Colmez Conjecture 4.2 Shimura Curves and Averaged Colmez Conjecture
  - 4.2.1 The Proof of Theorem 4.21 4.2.1 The Proof of Theorem 4.21
    - Decomposition of the Faltings height. Decomposition of the Faltings height.
    - Nearby pairs of CM types. Nearby pairs of CM types.
    - The Shimura curves X, X′, and X″. The Shimura curves X, X′, and X″.
    - Kodaira-Spencer isomorphisms. Kodaira-Spencer isomorphisms.
  - 4.2.2 Notation and Terminology 4.2.2 Notation and Terminology
  - 4.2.3 Decomposition of the Faltings Height 4.2.3 Decomposition of the Faltings Height
    - The τ-quotient space W(A, τ). The τ-quotient space W(A, τ).
    - The metric on the τ-component $N (A, τ)$ ⁠. The metric on the τ-component $N (A, τ)$ ⁠.
    - The τ-part of the Faltings height. The τ-part of the Faltings height.
    - The stable τ-part of the Faltings height. The stable τ-part of the Faltings height.
    - The decomposition of the Faltings height of Φ. The decomposition of the Faltings height of Φ.
  - 4.2.4 The Faltings Height of Nearby Pairs of CM Types 4.2.4 The Faltings Height of Nearby Pairs of CM Types
    - Nearby pairs of CM types. Nearby pairs of CM types.
    - Writing h(Φ₁, Φ₂) as $\frac{1}{2} h (A_{0}, τ)$ for some A₀ and τ. Writing h(Φ₁, Φ₂) as $\frac{1}{2} h (A_{0}, τ)$ for some A₀ and τ.
    - The construction of A₀. The construction of A₀.
  - 4.2.5 A Shimura Curve of PEL-type (Φ ₁ + Φ ₂) 4.2.5 A Shimura Curve of PEL-type (Φ ₁ + Φ ₂)
    - The various moduli problems. The various moduli problems.
    - Constructing the PEL Shimura varieties of type Φ₁ + Φ₂. Constructing the PEL Shimura varieties of type Φ₁ + Φ₂.
    - Special case of the PEL Shimura variety of type Φ₁ + Φ₂. Special case of the PEL Shimura variety of type Φ₁ + Φ₂.
    - Local systems in the special case. Local systems in the special case.
  - 4.2.6 Quaternionic Shimura curve X 4.2.6 Quaternionic Shimura curve X
    - Constructing the quaternionic Shimura curve. Constructing the quaternionic Shimura curve.
    - The integral model. The integral model.
    - Local Kodaira-Spencer map. Local Kodaira-Spencer map.
    - Arithmetic Hodge bundle. Arithmetic Hodge bundle.
  - 4.2.7 Link Between X′ and X 4.2.7 Link Between X′ and X
    - The Shimura curve X″. The Shimura curve X″.
    - Relating X′ and X″. Relating X′ and X″.
    - Relating X and X″. Relating X and X″.
    - Integral models. Integral models.
    - p-divisible groups. p-divisible groups.
  - 4.2.8 Proof of Theorem 4.21 4.2.8 Proof of Theorem 4.21
- 4.3 The Generalized Chowla-Selberg Formula 4.3 The Generalized Chowla-Selberg Formula
  - 4.3.1 Strategy of Proof of Theorem 4.25 4.3.1 Strategy of Proof of Theorem 4.25
    - Classical Lerch-Chowla-Selberg formula. Classical Lerch-Chowla-Selberg formula.
    - Gross-Zagier formula. Gross-Zagier formula.
    - Proof of Theorem 4.25 via Yuan-Zhang-Zhang. Proof of Theorem 4.25 via Yuan-Zhang-Zhang.
  - 4.3.2 Classical Lerch-Chowla-Selberg Formula 4.3.2 Classical Lerch-Chowla-Selberg Formula
    - Colmez conjecture in dimension one. Colmez conjecture in dimension one.
    - Faltings height of elliptic curves. Faltings height of elliptic curves.
    - Classical Lerch-Chowla-Selberg formula. Classical Lerch-Chowla-Selberg formula.
    - Obstructions to generalizing the method. Obstructions to generalizing the method.
  - 4.3.3 Gross-Zagier Formula 4.3.3 Gross-Zagier Formula
    - Modular forms and modular curves. Modular forms and modular curves.
    - The Gross-Zagier formula. The Gross-Zagier formula.
    - Strategy of proof. Strategy of proof.
  - 4.3.4 Proof of Theorem 4.25 4.3.4 Proof of Theorem 4.25
    - Proof of the fundamental identity. Proof of the fundamental identity.
    - Deduction of the generalized Chowla-Selberg formula. Deduction of the generalized Chowla-Selberg formula.
- 4.4 Higher Chowla-Selberg/Gross-Zagier Formula 4.4 Higher Chowla-Selberg/Gross-Zagier Formula
  - 4.4.1 Strategy of Proof of Theorem 4.32 4.4.1 Strategy of Proof of Theorem 4.32
    - Geometry of the moduli spaces of shtuka. Geometry of the moduli spaces of shtuka.
    - Relative trace formulae. Relative trace formulae.
  - 4.4.2 The Moduli Stack of Shtuka 4.4.2 The Moduli Stack of Shtuka
    - The stack of vector bundles, ${Bun}_{n}^{μ}$ ⁠. The stack of vector bundles, ${Bun}_{n}^{μ}$ ⁠.
    - The Hecke stack, ${Hk}_{n}^{μ}$ ⁠. The Hecke stack, ${Hk}_{n}^{μ}$ ⁠.
    - Moduli of shtuka for GL_n, ${Sht}_{n}^{μ}$ ⁠. Moduli of shtuka for GL_n, ${Sht}_{n}^{μ}$ ⁠.
    - Moduli of shtuka for PGL_n. Moduli of shtuka for PGL_n.
    - Hecke correspondences on PGL₂-shtuka. Hecke correspondences on PGL₂-shtuka.
    - Heegner-Drinfeld cycles. Heegner-Drinfeld cycles.
  - 4.4.3 The Relative Trace Formulae 4.4.3 The Relative Trace Formulae
    - The Eisenstein ideal. The Eisenstein ideal.
    - The analytic relative trace formula. The analytic relative trace formula.
    - Relation to L-functions. Relation to L-functions.
    - Spectral decomposition of intersection numbers. Spectral decomposition of intersection numbers.
    - The key identity. The key identity.
- Bibliography Bibliography
  - Notes
- Notes
- Notes
- List of Contributors
  - Notes
  - Notes

Chapter

Four Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang

Get access

https://doi.org/10.23943/princeton/9780691193779.003.0004

Pages

102–174

Published:

October 2019

Abstract

This chapter explores Shou-Wu Zhang's minicourse on Faltings heights and L-functions. It essentially consists of three parts. The first part discusses conjectures and results in the literature which give bounds, or formulae in terms of L-functions, for “Faltings heights.” The authors also mention various applications of such conjectures and results. The second part is devoted to the work of Yuan–Zhang in which they proved the averaged Colmez conjecture. Here, the authors detail the main ideas and concepts used in their proof. The third part focuses on the work of Yuan–Zhang in the function field world. Therein they compute special values of higher derivatives of certain automorphic L-functions in terms of self-intersection numbers of Drinfeld–Heegner cycles on the moduli stack of shtukas. The result of Yuan–Zhang might be viewed as a higher Gross–Zagier/Chowla–Selberg formula in the function field setting. The authors then motivate and explain the philosophy that Chowla–Selberg type formulae are special cases of Gross–Zagier type formulae coming from identities between geometric and analytic kernels.

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Contents

Four Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang

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