"Gross-Zagier" reading seminar 2024

SYNOPSIS

The seminar will meet Monday 3:30 - 5:00 pm in Van Vleck B321 .

We are planing to read Bruinier-Yang's "Faltings heights of CM cycles and derivatives of L-functions", with the main focus on small dimensional case, and see how their conjecture implies the Gross-Zagier formula.

There are double quotes in the title because our ultimate goal is not the proof of Gross-Zagier formula! Instead of understanding the detail of the proof, we prefer to understand the interesting contents presented in this paper.

To receive the messages, please subscribe the google group "2024_spring_gross-zagier@g-groups.wisc.edu".

References

• Jan Hendrik Bruinier and Tonghai Yang, Faltings heights of CM cycles and derivatives of L-functions

• Stephen S. Kudla , Notes on the local theta correspondence
• Wee Teck Gan, Automorphic Forms and the Theta Correspondence
• Dipendra Prasad , Weil Representation, Howe Duality, and the Theta Correspondence
• Haruzo Hida, Siegel-Weil formulas
• Jan Hendrik Bruinier and Tonghai Yang, Faltings heights of CM cycles and derivatives of L-functions, Section 2

• Jan Hendrik Bruinier, Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors, Section 2.2
• Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts
• Stephen S. Kudla, Integrals of Borcherds Forms
• Kathrin Bringmann and Stephen Kudla A classification of harmonic Maass forms
• Jan Hendrik Bruinier and Tonghai Yang, Faltings heights of CM cycles and derivatives of L-functions, Section 3 & 4

• Stephen S. Kudla, Algebraic cycles on Shimura varieties of orthogonal type
• Henri Gillet and Christophe Soulé, Arithmetic intersection theory
• Christophe Soulé, Arithmetic Intersection
• Jan Hendrik Bruinier and Tonghai Yang, Faltings heights of CM cycles and derivatives of L-functions, Section 6, 7 & 8

• J-P. Serre and H. M. Stark , Modular forms of weight 1/2
• Kevin Buzzard , Notes on modular forms of half-integral weight.

• Fabrizio Andreatta, Eyal Z. Goren, Benjamin Howard and Keerthi Madapusi Pera Height pairings on orthogonal Shimura varieties
• Stephen S. Kudla , From Modular Forms to Automorphic Representations
• Daniel Bump Automorphic Forms and Representations
• David E. Rohrlich, Modular forms lecture notes (unpublished)

Schedule

Mondays 3:30 - 5:00 pm, Van Vleck B321

Outline

Classical theta lifting

• (Ryan) Local and global aspects of Weil representation, and how to construct the theta series.
• (Jiaqi) The dual pair, theta lifting with the main focus on SL2 with O(0,2), O(1,2), and O(2,2).
• (Yu) Understand the definition of Siegel Eisenstein series, and the statement of the Siegel-Weil formula.

Geometry and arithmetic of Shimura varieties

• (Kevin) Definition of Shimura varieties, with the main focus on n=0, n=1 modular curve, and n=2 case.
• (Kevin) How to construct special cycles in the orthogonal Shimura varieties, in paritcular in those examples.
• (Yu) Arithmetic intersection theory.

Regularized theta lifting

• (Alejo) Definition and properties of harmonic weak Maass form, and its relation with weakly holomorphic modular form, understand the exact sequence between them
• (Jiaqi) Regularized theta lift for harmonic weak Maass forms
• (*)Integral of Brocherds forms (only known for weakly holomorhpic modualr form, but it would be interesting if you do for harmonic weak Maass form)

CM value of the regularized theta integral

• (Yu) Understand the proof of [Theorem 4.7, BY09]
• (Yu) Understand the statement of [Theorem 4.2, BY09]
• (Yu) How the arithmetic intersection theory enters into the story, and lead to the main conjecture.

Shimura lifting and the Gross-Zagier formula

• Definition of the half-weight integral, and the Shimura lift
• (Ryan) Gross-Zagier formula
• (*) How Bruinier-Yang's conjecture implies the Gross-Zagier formula (there are some gaps in the paper)

This seminar is organized by Yu LUO and Tonghai Yang. This page is took from Brian Lawrence's reading seminar