Shouwu Zhang - Heights of Ceresa and Gross-Schoen Cycles

Shouwu Zhang - Heights of Ceresa and Gross-Schoen Cycles

by Shouwu Zhang

Formal & Physical Sciences Mathematics PBMathematicsPBHNumber theory
🎙 Shouwu Zhang 👥 79K 📅 September 17, 2025 ⏱ 67 min 👁 751 📄 research talk 🧭 2026-08-03
Available in: English (current) Français

Keywords

heightsCeresa cyclesGross-Schoen cyclestriple product L-seriesarithmetic geometry

Summary

Shouwu Zhang, professor at Princeton University, gives a survey talk at IHES on recent work concerning the heights of Gross-Schoen and Ceresa cycles, and their connection to triple product L-series. He begins with the classical Abel-Jacobi map from a Riemann surface to its Jacobian, then discusses the arithmetic analogue developed by Weil and Neron. He introduces the notion of algebraic cycles and the Griffiths and Weil intermediate Jacobians. He recalls the work of Ceresa in the 1980s showing that the Ceresa cycle is not algebraically trivial for generic curves. He then presents the Gross-Schoen cycle, defined via a combination of diagonals in the triple product of a curve, and explains its relation to the triple product L-function. Zhang discusses his joint work with students on bounding the height of the Gross-Schoen cycle, relating it to the self-intersection of the relative dualizing sheaf and other invariants. He mentions applications to the Bogomolov conjecture and the effective Mordell conjecture. The talk is highly technical and aimed at researchers in arithmetic geometry.

169 words

Critical Evaluation

The talk by Shouwu Zhang is a high-level survey of recent developments in arithmetic geometry, focusing on the heights of algebraic cycles, specifically the Ceresa and Gross-Schoen cycles. Zhang is a leading expert in the field, and his presentation reflects deep knowledge and authority. The content is rigorous and well-structured, tracing the historical development from Abel-Jacobi theory to modern height theory and conjectures. The talk is dense with technical details, including definitions, theorems, and conjectures, and assumes a strong background in algebraic geometry and number theory. The speaker does not provide detailed proofs but gives an overview of the main ideas and results. The sources cited are primarily the works of Ceresa, Gross, Schoen, and others, which are foundational in this area. The talk is not intended for a general audience but for specialists. The title accurately reflects the content. The main strength is the clarity of exposition of complex ideas and the connection between different areas. However, the talk is quite technical, and some parts may be difficult to follow without prior knowledge. The lack of visual aids or written formulas on slides might hinder comprehension. Overall, this is a valuable resource for researchers in arithmetic geometry, providing insights into current research directions and open problems.

207 words

Title / Content Match

Le titre reflète bien le contenu : la conférence porte sur les hauteurs des cycles de Ceresa et de Gross-Schoen, comme annoncé.

Quality & Reliability

8/10

Talk by a leading expert (Shouwu Zhang) at IHES, presenting recent research in arithmetic geometry. The content is technical and based on established theories (heights, cycles, L-series). The talk is a survey of recent work, not a peer-reviewed publication, but the speaker's authority and the institutional setting lend high credibility.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Concurring Sources

  • Gross-Schoen cycle — Provides background on the cycle discussed in the talk.
  • Ceresa cycle — Provides background on the Ceresa cycle.

Contribution & Novelties

The talk provides a comprehensive survey of recent developments in the study of heights of algebraic cycles, particularly the Ceresa and Gross-Schoen cycles, and their connections to L-series. It highlights the speaker’s own contributions and those of his collaborators, including new inequalities and applications to conjectures like Bogomolov and effective Mordell. The talk synthesizes a large body of work and points to open problems, making it a valuable resource for researchers.

Pour aller plus loin :

  • Gross-Schoen cycle — Wikipedia article providing an overview.
  • Ceresa cycle — Wikipedia article on the Ceresa cycle.
  • Triple product L-function — Wikipedia article on triple product L-functions.

103 words

Radar Profile

The radar profile shows high scores in all dimensions, with particularly strong technical level and information quality. This reflects a dense, expert-level talk with substantial content and high reliability, though accessibility is limited to specialists.

Reliability 8/10