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arXiv · 2504.12131

Equidistribution of CM points on Shimura Curves and ternary theta series

Francesco Maria Saettone
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Abstract

We prove an equidistribution statement for the reduction of Galois orbits of CM points on the special fiber of a Shimura curve over a totally real field, considering both the split and the ramified case. The main novelty of the ramified case consists in the use of the moduli interpretation of the Cerednik--Drinfeld uniformisation. Our result is achieved by associating to the reduction of CM points certain Hilbert modular forms of weight $3/2$ and by analyzing their Fourier coefficients. Moreover, we also deduce the Shimura curves case of the integral version of the Andr\'e--Oort conjecture.

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Francesco Maria Saettone. 2025-04-16. Equidistribution of CM points on Shimura Curves and ternary theta series. https://arxiv.org/abs/2504.12131

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