The André-Oort conjecture via o-minimality (Q2825771)

The paper under review is of expository character. Its purpose is to explain the Pila-Zannier strategy for proving the André-Oort conjecture. The André-Oort conjecture is an important problem in arithmetic geometry concerning subvarieties of Shimura varieties. It attempts to characterize those varieties \(V\) for which the special points lying on \(V\) constitute a Zariski dense subset.NEWLINENEWLINEThe Pila-Zannier strategy for proving the conjecture relies on the Pila-Wilkie counting theorem on o-minimal structures and it is divided into two steps. Both of them require the hyperbolic Ax-Lindemann-Weiertrass conjecture, a geometric statement itself amenable to proof via o-minimality.NEWLINENEWLINEThe conjecture was first proven in the cocompact case by \textit{E. Ullmo} and \textit{A. Yafaev} [Duke Math. J. 163, No. 2, 433--463 (2014; Zbl 1375.14096)], for principally polarised abelian varieties by \textit{J. Pila} and \textit{J. Tsimerman} [Ann. Math. (2) 179, No. 2, 659--681 (2014; Zbl 1305.14020)] and finally by Klinger, Ullmo and Yafaev [\textit{B. Klingler} et al., Publ. Math., Inst. Hautes Étud. Sci. 123, 333--360 (2016; Zbl 1372.14016)].NEWLINENEWLINEThe paper does not mean to be a full treatment of the topic but a complete preparatory guide for graduate students. It mainly follows \textit{J. Milne} notes [``Introduction to Shimura varieties'', Preprint, \url{http://www.jmilne.org/math/xnotes/svi.pdf}] and \textit{E. Ullmo} paper [Compos. Math. 150, No. 2, 175--190 (2014; Zbl 1326.14062)].NEWLINENEWLINEFor the entire collection see [Zbl 1326.11003].