28 research outputs found
Hyperbolic Ax-Lindemann theorem in the cocompact case
We prove an analogue of the classical Ax-Lindemann theorem in the context of
compact Shimura varieties. Our work is motivated by J. Pila's strategy for
proving the Andr\'e-Oort conjecture unconditionallyComment: To appear in Duke Mathematical Journa
Proving the triviality of rational points on Atkin-Lehner quotients of Shimura curves
In this paper we give a method for studying global rational points on certain
quotients of Shimura curves by Atkin-Lehner involutions. We obtain explicit
conditions on such quotients for rational points to be ``trivial'' (coming from
CM points only) and exhibit an explicit infinite family of such quotients
satisfying these conditions.Comment: 25 pages. To appear in Mathematische Annale
An unconditional proof of the Andre-Oort conjecture for Hilbert modular surfaces
We give an unconditional proof of the André–Oort conjecture for Hilbert modular surfaces asserting that an algebraic curve contained in such a surface and containing an infinite set of special points, is special. The proof relies on a combination of Galois-theoretic techniques and results from the theory of o-minimal structures
Height functions on Hecke orbits and the generalised Andr\'e-Pink-Zannier conjecture
We introduce and study the notion of a generalised Hecke orbit in a Shimura
variety. We define a height function on such an orbit and study its properties.
We obtain a lower bounds for the size of Galois orbits of points in a
generalised Hecke orbit in terms of these height, assuming a version of the
Mumford-Tate conjecture. We then use it to prove the generalised
Andr\'e-Pink-Zannier conjecture under this assumption by implementing the
Pila-Zannier strategy