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8 reviews1. IntroductionThe central object of study in this article is the following conjecture.Conjecture 1.1 (André-Pink-Zannier). — Let S be a Shimura variety and a subset of ageneralised Hecke orbit in S (as in [40, Def. 2.1]). Then the irreducible components of the Zariskiclosure of are weakly special subvarieties.This conjecture is an important special case of the Zilber-Pink conjectures forShimura varieties, which has recently been and continues to be a subject of active research.A special case of Conjecture 1.1 was first formulated in 1989 by Y. André in [1,§X 4.5, p. 216 (Problem 3)]. Conjecture 1.1 was then stated in the introduction to thesecond author’s 2000 PhD thesis [57],1 following discussions with Bas Edixhoven. Bothstatements refer to classical Hecke orbits, rather than generalised Hecke orbits (cf. [40,§2.5.1]).1 The statement there uses the terminology ‘tothad proved in [30] that the two notions are equivalent.