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

DOI: 10.2140/pjm.2019.303.165

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the number of principally polarized abelian varieties over $k$), but the symplectic group in the Kottwitz formula has been replaced by the general linear group.

Keywords

ordinary abelian variety, locally symmetric space, Kottwitz formula

Mathematical Subject Classification 2010

Primary: 11F99, 11G25, 14G35, 14K10

Milestones

Received: 22 August 2018

Revised: 3 April 2019

Accepted: 4 April 2019

Published: 21 December 2019