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arXiv · 2403.18688

The Gross--Kohnen--Zagier theorem via $p$-adic uniformization

Lea Beneish·Henri Darmon·Lennart Gehrmann·Martí Roset
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Abstract

This article gives a new proof of the Gross--Kohnen--Zagier theorem for Shimura curves which exploits the $p$-adic uniformization of Cerednik--Drinfeld. The explicit description of CM points via this uniformization leads to an expression relating the Gross--Kohnen--Zagier generating series to the ordinary projection of the first derivative, with respect to a weight variable, of a $p$-adic family of positive definite ternary theta series.

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Lea Beneish, Henri Darmon, Lennart Gehrmann, Martí Roset. 2024-03-27. The Gross--Kohnen--Zagier theorem via $p$-adic uniformization. https://arxiv.org/abs/2403.18688

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