Simple mass formulas on Shimura varieties of PEL-type

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 $\Q$-group $G$ relative to an open compact subgroup $U$ of $G(\A_f)$, or simply called an {ıt arithmetic mass}. We classify the special objects f...

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Bibliographic Details
Main Author Yu, Chia-Fu
Format Journal Article
LanguageEnglish
Published 18.03.2006
Subjects
Mathematics - Number Theory
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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 $\Q$-group $G$ relative to an open compact subgroup $U$ of $G(\A_f)$, or simply called an {ıt arithmetic mass}. We classify the special objects for which our formulation remains valid over algebraic 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.
DOI:10.48550/arxiv.math/0603451