Xinyi Yuan - A Quantitative Version of the Uniform Mordell Conjecture

Xinyi Yuan - A Quantitative Version of the Uniform Mordell Conjecture

by Xinyi Yuan

Formal & Physical Sciences Mathematics PBMathematicsPBHNumber theory
🎙 Xinyi Yuan 👥 79K 📅 September 17, 2025 ⏱ 65 min 👁 1K 📄 research talk 🧭 2026-08-03
Available in: English (current) Français

Keywords

Mordell conjectureuniform boundVojta's inequalityFaltings heightJacobian rank

Summary

Xinyi Yuan presents a quantitative version of the uniform Mordell conjecture, a result providing explicit upper bounds on the number of rational points on a smooth projective curve of genus >1 over a number field. The talk begins by reviewing the Mordell conjecture (Faltings 1983) and its three proofs: Faltings’ original proof using Faltings heights and p-adic Hodge theory, Vojta’s proof via Diophantine approximation and arithmetic Riemann-Roch, and the recent proof by Lawrence-Venkatesh using periodic Hodge theory. Yuan then introduces the uniform bound by Dimitrov-Gao-Habegger and Kuhne, which depends on the genus and the Mordell-Weil rank of the Jacobian. The main result, joint work with Jiawei Yu and Shengxuan Zhou, gives explicit constants: the number of rational points is bounded by C1(G) * C2(G)^R, where C1(G) is a ‘Manin-Mumford constant’ and C2(G) is a ‘Vojta constant’ that tends to 1 as G→∞. The proof combines bounds for points of large height (via Vojta’s inequality) and small height (via equidistribution and Bogomolov-type results). A corollary is presented: for hyperelliptic curves over Q of genus >16, the average number of rational points, ordered by height, is bounded by 3, using the average size of 2-Selmer groups (Bhargava-Gross). The talk concludes with remarks on potential improvements and connections to the effective Mordell conjecture and the ABC conjecture.

214 words

Critical Evaluation

The talk is a high-level research presentation aimed at specialists in arithmetic geometry. The speaker, Xinyi Yuan, is a renowned mathematician, and the content is based on rigorous mathematical work, likely to be published in a top journal. The presentation is logically structured: it starts with background, reviews existing results, and then presents the new theorem with its proof strategy. The mathematical arguments are solid, relying on established techniques such as Vojta’s inequality, equidistribution, and arithmetic intersection theory. The explicit bounds, though large, represent a significant step towards an effective version of the Mordell conjecture. The talk does not provide full details of the proofs, but that is expected in a seminar format. The sources cited are appropriate and include key references in the field. The title accurately reflects the content. The main limitation is the technical level, which makes it inaccessible to non-experts, but this does not detract from the scientific value. The talk also includes a corollary about average number of rational points on hyperelliptic curves, which is an interesting application. Overall, the talk is of high quality and contributes to the field by providing explicit uniform bounds.

190 words

Title / Content Match

The title accurately reflects the content: the talk presents a quantitative version of the uniform Mordell conjecture, with explicit bounds.

Quality & Reliability

8/10

Talk by a leading expert in arithmetic geometry, presenting joint work with Jiawei Yu and Shengxuan Zhou. The content is technical and rigorous, with references to established theorems (Faltings, Vojta, Dimitrov-Gao-Habegger, Kuhne). However, the talk is not peer-reviewed and some details are omitted for time.

Key Moments

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

Cited Sources

  • Carmin.tv — Video platform hosting the talk

Concurring Sources

  • Carmin.tv — Platform hosting the talk, likely to have related content

Contribution & Novelties

The talk presents a new quantitative version of the uniform Mordell conjecture, providing explicit bounds on the number of rational points on curves of genus >1. This is a significant improvement over previous non-effective results, as it gives concrete constants depending only on the genus. The work combines techniques from Vojta’s inequality, equidistribution, and arithmetic intersection theory, and also yields a corollary on the average number of rational points for hyperelliptic curves.

Pour aller plus loin :

115 words

Radar Profile

The radar profile shows high scores in quantitative information, qualitative information, and technical level, reflecting a dense and rigorous research talk. The reliability score is slightly lower due to the lack of peer review and the informal nature of a seminar presentation.

Reliability 8/10