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arXiv · 2210.16218

Zero-dimensional Shimura varieties and central derivatives of Eisenstein series

Siddarth Sankaran
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Abstract

We formulate and prove a version of the arithmetic Siegel-Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of ``special" divisors in terms of Green functions at archimedean and non-archimedean places, and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel-Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results in the literature.

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Siddarth Sankaran. 2022-10-28. Zero-dimensional Shimura varieties and central derivatives of Eisenstein series. https://arxiv.org/abs/2210.16218

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