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**Torsion in algebraic groups and problems which arise.**
*(English)*
Zbl 1540.11072

Hujdurović, Ademir (ed.) et al., European congress of mathematics. Proceedings of the 8th congress, 8ECM, Portorož, Slovenia, June 20–26, 2021. Berlin: European Mathematical Society (EMS). 161-180 (2023).

This entertaining introductory article starts at a very basic level and, without of course giving any proofs whatsoever, leads the reader in a few pages to some of the hardest problems in modern number theory. The first few pages set the scene by discussing torsion in \({\mathbb G}_a\), \({\mathbb G}_m\) (already we are discussing cyclotomic fields, which are only easy in the sense that the main problems were solved by Gauss and Galois), and elliptic curves (by now we have reached major theorems from the 1970s). The second section moves on to questions of algebraic relations among torsion points and the conjectures of Lang and of Manin-Mumford, proved by Sarnak and others and by Raynaud respectively, and to unlikely intersections and the work of Zilber and Pink before proceeding to the André-Oort conjecture. Section 3 concerns the problems that arise in families (by now the results being indicated are very recent); Section 4 states some wide-ranging applications, also very recent; Section 5 is a very short list of very hard open problems; and Section 6 is a brief apology for the reference list, because the author has given generous credit in the text to a large number of people, and has run out of space to list all their papers directly.

For the entire collection see [Zbl 1519.00033].

For the entire collection see [Zbl 1519.00033].

Reviewer: G. K. Sankaran (Bath)

### References:

[1] | The book [1] on Unlikely Intersections was written about 10 years ago: much work has appeared later, but the book contains an account of a substantial part of the contents of these notes, and many references. |

[2] | The more recent survey paper [2] contains further descriptions and more updated bibliography with respect to the former reference. |

[3] | U. Zannier, Some Problems of Unlikely Intersections in Arithmetic and Geometry. Ann. of Math. Stud. 181, Princeton University Press, Princeton, NJ, 2012 Zbl 1246.14003 MR 2918151 · Zbl 1246.14003 |

[4] | U. Zannier, Some specialization theorems for families of abelian varieties. Münster J. Math. 13 (2020), no. 2, 597-619 Zbl 1455.14087 MR 4130694 · Zbl 1455.14087 |

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