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# Congruence Relations for Shimura Varieties Associated with $GU(n-1,1)$

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Published:2013-11-13
Printed: Dec 2014
• Jean-Stefan Koskivirta,
Strasbourg University
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## Abstract

We prove the congruence relation for the mod-$p$ reduction of Shimura varieties associated to a unitary similitude group $GU(n-1,1)$ over $\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$ is even was obtained by T. Wedhorn and O. B?ltel, as a special case of a result of B. Moonen, when the $\mu$-ordinary locus of the $p$-isogeny space is dense. This condition fails in our case. We show that every supersingular irreducible component of the special fiber of $p\textrm{-}\mathscr{I}sog$ is annihilated by a degree one polynomial in the Frobenius element $F$, which implies the congruence relation.
 Keywords: Shimura varieties, congruence relation
 MSC Classifications: 11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G35 - Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 14K10 - Algebraic moduli, classification [See also 11G15]

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