Vol. 303, No. 1, 2019

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Real structures on polarized Dieudonné modules

Mark Goresky and Yung sheng Tai

Vol. 303 (2019), No. 1, 217–241

We define an “antiholomorphic involution” of a module M over the Dieudonné ring (k) of a finite field k with q = pa elements to be an involution τ : M M that switches the action of a with that of Va. The definition extends to include quasi-polarizations of Dieudonné modules. Nontrivial examples exist. The number of isomorphism classes of quasi-polarized Dieudonné modules within a fixed isogeny class is shown to be given by a twisted orbital integral over the general linear group. Earlier (Pacific J. Math. 303:1 (2019), 165–215) we considered these notions in the case of ordinary abelian varieties over k, in which case the contribution at p to the number of isomorphism classes within an isogeny class was shown to be given by an ordinary orbital integral over the general linear group. The definitions here are shown to be equivalent to those in our previous paper and, as a consequence, the equality of the orbital integrals of both types is proven.

Dieudonné module, abelian variety, real structure
Mathematical Subject Classification 2010
Primary: 14G35, 16W10, 14K10, 22E27
Received: 22 August 2018
Revised: 27 October 2018
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