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arXiv · 2309.08886

Heights of Special Points on Quaternionic Shimura Varieties

Roy Zhao
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Abstract

Let $B/F$ be a quaternion algebra over a totally real number field. We give an explicit formula for heights of special points on the quaternionic Shimura variety associated with $B$ in terms of Faltings heights of CM abelian varieties. Special points correspond to CM fields $E$ and partial CM-types $\phi \subset \mathrm{Hom}(E, \mathbb{C})$. We then show that our height is compatible with the canonical height of a partial CM-type defined by Pila, Shankar, and Tsimerman. This gives another proof that the height of a partial CM-type is bounded subpolynomially in terms of the discriminant of $E$.

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Roy Zhao. 2023-09-16. Heights of Special Points on Quaternionic Shimura Varieties. https://arxiv.org/abs/2309.08886

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