Abstract
Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the centre of the endomorphism ring of the corresponding \(\mathbb {Z}\)-Hodge structure. Our bound is the final step needed to complete a proof of the André–Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier.
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Acknowledgments
Both authors would like to thank Emmanuel Ullmo and Andrei Yafaev for suggesting that they work together on this problem and for numerous conversations regarding the subject of this paper. The authors also owe a special thank you to Philipp Habegger who suggested the use of Chow polynomials to prove Proposition 3.11. They are grateful to Ziyang Gao for pointing out the issue with quantifiers in Theorem 1.4 which is needed for it to imply Theorem 1.2. Both authors would like to thank the referee for their reading of the manuscript and their helpful comments. The first author is indebted to the Engineering and Physical Sciences Research Council and the Institut des Hautes Études Scientifiques for their financial support. The second author was funded by European Research Council Grant 307364 “Some problems in Geometry of Shimura varieties”.
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Daw, C., Orr, M. Heights of pre-special points of Shimura varieties. Math. Ann. 365, 1305–1357 (2016). https://doi.org/10.1007/s00208-015-1328-3
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DOI: https://doi.org/10.1007/s00208-015-1328-3