ICM 2026 Invited Section Lecture - Jennifer Balakrishnan

ICM 2026 Invited Section Lecture - Jennifer Balakrishnan

by Jennifer Balakrishnan · View the channel on PSI

Formal & Physical Sciences Mathematics PBMathematicsPBHNumber theory
🎙 Jennifer Balakrishnan 👥 61K 📅 September 28, 2026 ⏱ 39 min 👁 487 📄 literature review 🧭 2026-10-06
Available in: English (current) Français

Keywords

p-adic integrationSelmer varietiesChabauty's theoremColeman integrationMordell-Weil rank

Summary

Jennifer Balakrishnan gives an invited lecture on explicit p-adic methods for determining rational points on curves of genus at least 2. She begins with the motivating example of the non-split Cartan modular curve of level 19, whose rational points were recently computed. The historical background starts with Faltings’ finiteness theorem and Chabauty’s 1941 method, which applies when the rank of the Jacobian is less than the genus. Coleman’s p-adic integration made this explicit, yielding bounds on rational points. Kim later reframed the method using unipotent fundamental groups and conjectured that these sets eventually cut out the rational points exactly. The depth-2 case, quadratic Chabauty, has been made explicit and implemented in the QCMod Magma package, leading to many successful computations on modular curves. The speaker details the algorithm’s components, including p-adic heights, Hecke correspondences, and local height computations, and discusses computational bottlenecks and recent work on non-split 19. The talk also mentions mock rational points and broader applications to Serre uniformity.

161 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a high-value synthesis of a sophisticated area of arithmetic geometry, tracing the development from classical Chabauty to modern quadratic Chabauty and its explicit implementation. The argumentation is clear: each method is motivated by a specific example, the hypotheses are carefully stated, and the logical flow from theory to computation is compelling. The speaker highlights both successes and unresolved computational challenges, giving a balanced view of the field’s capabilities and limits.

Scientific Rigor, Source Quality, Title Accuracy

The presentation is rigorous, with precise statements of rank conditions and hypotheses. The speaker cites key papers (Faltings, Chabauty, Coleman, Kim, Balakrishnan-Dogra-Muller-Tuitman-Vonk) and mentions the LMFDB as a data source, though no explicit hyperlinks are provided. The content is consistent with published literature. The title is generic but appropriate for an invited lecture; the specific subject is conveyed in the description.

149 words

Title / Content Match

The title accurately identifies this as an invited lecture at ICM 2026 and names the speaker; the specific topic is stated in the description, which aligns with the content presented.

Quality & Reliability

8/10

The talk is a rigorous survey presented by an established researcher, with clear references to foundational theorems (Faltings, Chabauty, Coleman, Kim) and recent explicit results. The content is mathematically sound and well-structured, though it relies heavily on prior knowledge and does not provide full proofs.

Key Moments

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

Cited Sources

Contribution & Novelties

The talk provides a state-of-the-art overview of explicit p-adic methods, particularly focusing on quadratic Chabauty and its recent successful applications, including the resolution of rational points on non-split Cartan modular curves. It highlights the computational challenges and the importance of model choices and algorithmic efficiency. The speaker also discusses mock rational points, an open area of research.

Pour aller plus loin :

  • LMFDB — Database used for modular curve data and plane models.
  • Mordell-Weil theorem — Relevant to the rank of Jacobian; see Wikipedia article .
  • Kim’s unipotent fundamental group approach — Concept without URL; foundational to the depth-2 quadratic Chabauty.
  • Tuitman’s algorithm for computing Coleman integrals — Concept without URL; key to the explicit computations.

116 words

Radar Profile

The profile shows very high scores in quality, technical level, and reliability, with a slightly lower but still strong score in quantity, reflecting a dense but focused lecture that covers the essential material comprehensively.

Reliability 8/10