The André-Oort conjecture via o-minimality [chapter]

Christopher Daw, G. O. Jones, A. J. Wilkie
O-Minimality and Diophantine Geometry  
1 the case of Abelian varieties (respectively algebraic tori), where special subvarieties are the translates of Abelian subvarieties (respectively subtori) by torsion points. A key property of special subvarieties is that connected components of their intersections are themselves special subvarieties. Thus, any subvariety Y of S is contained in a smallest special subvariety. If this happens to be a connected component of S itself, then we say that Y is Hodge generic in S. We refer to the
more » ... refer to the special subvarieties of dimension zero as special points. Special subvarieties contain a Zariski (in fact, analytically) dense set of special points. The André-Oort conjecture predicts that this property characterises special subvarieties: Conjecture 1.1 (André-Oort) Let S be a Shimura variety and let Σ be a set of special points contained in S. Every irreducible component of the Zariski closure of ∪ s∈Σ s in S is a special subvariety. A connected component of S arises as a quotient Γ\D, where D is a certain type of complex manifold called a Hermitian symmetric domain, and Γ is a certain type of discrete subgroup of Hol(D) + called a congruence subgroup. From now on, we will use S to denote this component. By [10], §3, there exists a semi-algebraic fundamental domain F ⊂ D for the action of Γ. By [10], Theorem 1.2, when the uniformisation map π : D → S is restricted to F, one obtains a function definable in the o-minimal structure R an,exp . Through these observations, the André-Oort conjecture becomes amenable to tools from o-minimality.
doi:10.1017/cbo9781316106839.006 fatcat:fbdwpw2lnrdajazgrnwd7f3pwa