Abstract
We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of
A Appendix
We prove here Theorem 7.6 when
Case (i): E = A
In this case,
It is clear that for all
Let
and
with
By (A.1),
Integrating both side with respect to
Let now
By Pila–Wilkie [64, Theorem 3.4], there exists an semi-algebraic block
and then
Case (ii): E = T
Define the norm of
It is clear that for all
Let
where
By (A.4),
Integrating both side with respect to
we have
by (A.5) and (A.6). Hence
Let now
By Pila–Wilkie [47, Theorem 3.6], there exists an semi-algebraic subset
and then
Acknowledgements
This topic was introduced to me by Emmanuel Ullmo. This first part of this paper (Sections 2–8) was done in Leiden University, while the two main theorems were proved when I was in Université Paris-Sud. I would like to express my gratitude to my supervisors Emmanuel Ullmo and Bas Edixhoven for weekly discussions and their valuable suggestions for the writing. I would like to thank Martin Orr for having pointed out a serious gap in Section 9 in a previous version as well as his several valuable remarks, especially for the last part of Section 10. Ya’acov Peterzil pointed out to me that the proof of the definability in Section 10.1 in a previous version was wrong. I have benefited a lot from the discussion with him and Sergei Starchenko for this definability problem. I also had some interesting discussion with Daniel Bertrand and Chao Zhang. Yves André, Daniel Bertrand, Bruno Klingler and Martin Orr have read a previous version of the manuscript and gave me some suggestions to improve the writing of both math and language. I would also like to thank them here. Finally, I thank the referee for his/her careful reading and helpful suggestions thanks to which this article has been improved.
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