Vol. 303, No. 1, 2019

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Ordinary points mod $p$ of GL$_n(\mathbb{R})$-locally symmetric spaces

Mark Goresky and Yung sheng Tai

Vol. 303 (2019), No. 1, 165–215

Locally symmetric spaces for GLn() 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 (T,F,V ) 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the number of principally polarized abelian varieties over k), but the symplectic group in the Kottwitz formula has been replaced by the general linear group.

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ordinary abelian variety, locally symmetric space, Kottwitz formula
Mathematical Subject Classification 2010
Primary: 11F99, 11G25, 14G35, 14K10
Received: 22 August 2018
Revised: 3 April 2019
Accepted: 4 April 2019
Published: 21 December 2019
Mark Goresky
School of Mathematics
Institute for Advanced Study
Princeton, NJ
United States
Yung sheng Tai
Department of Mathematics
Haverford College
Haverford, PA
United States