Arithmetic volumes of unitary Shimura varieties

Jan Hendrik Bruinier; Benjamin Howard

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arXiv · 2105.11274

Arithmetic volumes of unitary Shimura varieties

Jan Hendrik Bruinier·Benjamin Howard

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Abstract

The integral model of a $\mathrm{GU}(n-1,1)$ Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet $L$-functions. We also determine the arithmetic volumes of Kudla-Rapoport divisors and relate them to coefficients of Eisenstein series.

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Jan Hendrik Bruinier, Benjamin Howard. 2021-05-24. Arithmetic volumes of unitary Shimura varieties. https://arxiv.org/abs/2105.11274

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