Daw, CM; (2014) Around the André-Oort conjecture. Doctoral thesis , UCL (University College London).

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Abstract

In this thesis we study the André-Oort conjecture, which is a statement regarding subvarieties of Shimura varieties that contain a Zariski dense set of special points. In particular, we investigate two different strategies for proving the conjecture. The first is the so-called Pila-Zannier strategy, which is a striking application of the Pila-Wilkie counting theorem from o-minimality and has led to a number of unconditional proofs in special cases. We present one such proof here, for Hilbert modular surfaces, and also explain how the Pila-Zannier strategy generalises to all Shimura varieties. We subsequently exhibit a result on torsion in the class groups of algebraic tori, obtained from an investigation into some of the relevant arithmetic. The second strategy originated in the work of Edixhoven and ultimately led to a proof of the full conjecture under the generalised Riemann hypothesis by Klingler, Ullmo and Yafaev. However, this proof diverged from the original strategy of Edixhoven, which used only tools from arithmetic geometry, in that it also relied on ergodic theory. Here we explain how to eliminate ergodic theory from the proof, first in a special case and then in general, by introducing a new lower bound for the degrees of special subvarieties. First, however, we give an introduction to the theory of Shimura varieties for the purposes of studying this subject.

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