A Proof of the André - Oort Conjecture via Mathematical Logic

INTRODUCTION Extending work of Bombieri and Pila on counting lattice points on convex curves [3], Pila and Wilkie proved a strong counting theorem on the number of rational points in a more general class of sets definable in an o-minimal structure on the real numbers [37]. Following a strategy proposed by Zannier, the Pila-Wilkie upper bound has been leveraged against Galois-theoretic lower bounds in to prove theorems in diophantine geometry to the effect that for certain algebraic varieties the algebraic relations which may hold on its " special points " are exactly those coming from " special varieties ". Of these results, Pila's unconditional proof of the André-Oort conjecture for the j-line is arguably the most spectacular and will be the principal object of this resumé. Readers interested in a survey with more details about some of the other results along these lines, specifically the Pila-Zannier reproof of the Manin-Mumford conjecture and the Masser-Zannier theorem about simultaneous torsion in families of elliptic curves, may wish to consult my notes for the Current Events Bulletin lecture [43]. Acknowledgements. I wish to thank M. Aschenbrenner, J. Pila and U. Zannier for their advice and especially for suggesting improvements to this text.