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Simple mass formulas on Shimura varieties of PEL-type. (English) Zbl 1257.11059
Summary: We give a unified formulation of a mass for arbitrary abelian varieties with PEL-structures and show that it equals a weighted class number of a reductive \(\mathbb Q\)-group \(G\) relative to an open compact subgroup \(U\) of \(G(\mathbb A_{f})\), or simply called an arithmetic mass. We classify the special objects for which our formulation remains valid over algebraically closed fields. As a result, we show that the set of basic points in a mod \(p\) moduli space of PEL-type with a local condition (and a mild condition subject to the Hasse principle) can be expressed as a double coset space and its mass equals an arithmetic mass. The moduli space does not need to have good reduction at \(p\). This generalizes a well-known result for superspecial abelian varieties.

11G18 Arithmetic aspects of modular and Shimura varieties
11E41 Class numbers of quadratic and Hermitian forms
14G35 Modular and Shimura varieties
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