Abstract

Locally symmetric spaces for ${GL}_n\left(ℝ\right)$ parametrize polarized complex abelian varieties with real structure (antiholomorphic involution). We introduce a mod $p$ analog. We define an “antiholomorphic” involution (or “real structure”) on an ordinary abelian variety (defined over a finite field $k$) to be an involution of the associated Deligne module $\left(T,F,V\right)$ that exchanges $F$ (the Frobenius) with $V$ (the Verschiebung). The definition extends to include principal polarizations and level structures. We show there are finitely many isomorphism classes of such objects in each dimension, and give a formula for this number that resembles the Kottwitz “counting formula” (for the number of principally polarized abelian varieties over $k$), but the symplectic group in the Kottwitz formula has been replaced by the general linear group.

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Mathematical Subject Classification 2010

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