ICM 2026 Invited Section Lecture - Michael Stoll

Value of the Information & Strength of the Argument

The talk provides a high-value survey of the field, merging classical results with very recent advances. The argumentation is rigorous: each claim is either a proven theorem or explicitly identified as conditional or open. The presentation of explicit constants is particularly valuable, as it gives concrete, usable bounds that improve on previous non‑explicit estimates. The logical structure is clear, moving from the classical facts to modern uniform bounds, then to lower constructions and heuristic expectations. The speaker also carefully distinguishes between unconditional results and those depending on unproved conjectures (e.g., Lang’s conjecture, bound on ranks), which strengthens the credibility of the narrative.

Scientific Rigor, Source Quality, Title Accuracy

The speaker is a recognized expert, and the talk is part of an invited lecture at ICM 2026, ensuring an authoritative context. The content relies on published theorems (e.g., Faltings, Mordell–Weil, Chabauty) and recent preprints, which are correctly attributed. References to the work of others are precise, and the speaker notes when results are his own or joint work. The title of the video, however, is generic and does not convey the subject matter, which is an issue for discoverability but does not detract from the scientific content. The talk maintains high standards of rigor, with explicit hypotheses and precise constants.

The title is generic, giving only the conference and speaker name, but the description and content clearly address rational points on curves, which is not hinted at in the title. However, the title does not mislead.

9/10

Rigorous mathematical exposition by a leading expert, presenting precise theorems, explicit constants, and open problems with clear references to established results. No speculation or unsubstantiated claims.

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

• 01:00Introduction of the question: How many rational points can a curve have? Motivated by title borrowed from Caporaso–Harris–Mazur.
• 05:00Definition of nice curves and the three cases based on genus: genus 0, genus 1 (elliptic curves), genus ≥ 2 (Faltings theorem).
• 15:00Introduction of Jacobian and the Mordell–Weil theorem for abelian varieties; the embedding of curve into its Jacobian.
• 25:00Recent uniform bounds: explicit constants for the number of rational points as a function of genus and rank (Dimitrov–Gao–Habegger, Stoll et al.).
• 30:00Discussion of lower bounds (e.g., genus-2 record of 642 points) and expected bounds from unlikely intersections; open problems on ranks and optimal growth.