André-Oort conjecture for arithmetic varieties
Let be an arithmetic variety, let be an algebraic subvariety, and let be the set of special points of that belong to . André-Oort conjecture. The Zariski closure of is a finite union of special subvarieties of . The source states that this conjecture has been proved, so it is included as a resolved conjecture rather than an open problem.
References
Primary source
Rodolphe Richard and Andrei Yafaev, “A Common Generalisation of the André-Oort and André-Pink-Zannier Conjectures”, arXiv:2401.03528 (2024).
Additional references
20 papers in this index state this conjecture (1999–2024). The statement above is taken from the most recent of them; the others are arXiv:2306.02290, arXiv:2112.13814, arXiv:2103.15717, arXiv:1812.04341, arXiv:1711.09387, arXiv:1607.07843, arXiv:1506.01466, arXiv:1503.07332, arXiv:1502.00822, arXiv:1405.6053, arXiv:1209.0934, arXiv:1209.0939, and 7 more.
Progress summary
A claimed proof announced in 2021 and reported in 2022 says the conjecture is true in full generality, replacing earlier conditional results.
The conjecture asserts that special points in an algebraic subvariety of an arithmetic variety have Zariski closure equal to a finite union of special subvarieties.
Known results
- Klingler and Yafaev, 2014: proved the conjecture assuming the Generalized Riemann Hypothesis for CM fields.
- Klingler and Yafaev, 2014: proved it unconditionally when the relevant Mumford–Tate groups lie in one -conjugacy class.
2021–2024 unconditional proof claim
A September 2021 preprint claims an unconditional proof for every Shimura variety. On February 3, 2022, Quanta reported the result by Jonathan Pila, Ananth Shankar, and Jacob Tsimerman as removing the earlier conditionality; a January 2024 survey and a 2026 note describe André–Oort as proved. No retrieved source reports a counterexample, gap, withdrawal, or retraction.
Current status (as of September 2026): the conjecture is reported as unconditionally proved in full generality, but this automated record treats that resolution as unverified.
Sources
- arxiv.org
- annals.math.princeton.edu
- ar5iv.labs.arxiv.org
- arxiv.org
- quantamagazine.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- arxiv.org
- numdam.org
- arxiv.org
- mathoverflow.net
- openai.com
- cdn.openai.com
- cdn.openai.com
- ar5iv.labs.arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- quantamagazine.org
Solutions 0
No solutions have been posted yet.