Mathematics#16 in July 2026 EditionResolution of the uniform boundedness conjecture for torsion points on curves of genus two
Laurent Beaulieu, Anika Mehta, Christoph Schmidt et al.
OPENSTEM EDITORIAL SUMMARY
What the research examines
This monumental paper establishes that the number of torsion points lying on an embedded algebraic curve of genus two inside its Jacobian variety is bounded solely in terms of the genus, proving the strong uniform Manin-Mumford conjecture.
Why it may matter
Resolves a foundational 40-year open problem at the intersection of arithmetic geometry, height theory, and Diophantine equations.
What to examine carefully
Methods rely deeply on intersection theory on abelian schemes and do not immediately provide effective bounds for curves of genus g >= 3.
Editorial scope: This is a concise, independently written introduction based on the publisher information, not a full systematic review or independent replication. It must not replace reading the original methods, limitations and conclusions.
STUDENT RESEARCH LENS · ORIGINAL OPENSTEM NOTES
Try a small, reproducible follow-up
Compute canonical heights of torsion divisors on Jacobians of hyperelliptic curves over algebraic number fields using SageMath.
Questions to investigate
- How are Betti map degenerations controlled near the boundary of the moduli space?
- What quantitative upper bound on torsion point cardinality is explicitly extracted from the proof?
These are independently proposed educational investigations, not activities that the original research authors claim to have completed.
Publication & researcher information
JournalAnnals of Mathematics
Published online2026-07-02
Authors (selected)Laurent Beaulieu, Anika Mehta, Christoph Schmidt et al.
University / laboratorySorbonne Université & CNRS
CountryFrance
Multi-author research may involve universities in several countries. Unverified institutions and contact addresses are not inferred from authors' names.