Abstract
Let S be a Shimura variety. We conjecture that the heights of special points in \(S(\overline{\mathbb {Q}})\) are discriminant negligible with respect to some Weil height function \(h:S(\overline{\mathbb {Q}})\rightarrow \mathbb {R}\). Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In particular, we give a new proof of a theorem of Tsimerman on lower bounds for Galois degrees of special points in Shimura varieties of abelian type. This gives a new proof of the André–Oort conjecture for such varieties that avoids the use of Masser–Wüstholz isogeny estimates, replacing them by a point-counting argument.
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Acknowledgements
Schmidt thanks the EPSRC for support under grant EP/N007956/1. Schmidt thanks the Weizmann Institute for their hospitality and the University of Basel for its support. Yafaev is very grateful to the University of Manchester and Weizmann Institute for their hospitality and to Leverhulme Trust for support. We are very grateful to Jacob Tsimerman for pointing out a gap in the first version of the paper. The third author is very grateful to Emmanuel Ullmo and Rodolphe Richard for discussions related to the subject of this paper. The second author thanks Philipp Habegger for helpful discussions about heights. We are grateful to the referee for suggestions and comments.
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Dedicated to the memory of Bas Edixhoven.
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This research was supported by the ISRAEL SCIENCE FOUNDATION (Grant No. 1167/17) and by funding received from the MINERVA Stiftung with the funds from the BMBF of the Federal Republic of Germany. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No 802107) Yafaev was supported by a Leverhulme research grant RPG-2019-180. Schmidt was supported by the Engineering and Physical Sciences Research Council Grant EP/N007956/1.
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Binyamini, G., Schmidt, H. & Yafaev, A. Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach. Math. Ann. 385, 961–973 (2023). https://doi.org/10.1007/s00208-021-02309-0
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DOI: https://doi.org/10.1007/s00208-021-02309-0