#### 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
##### Abstract

We define an “antiholomorphic involution” of a module $M$ over the Dieudonné ring $\mathsc{ℰ}\left(k\right)$ of a finite field $k$ with $q={p}^{a}$ elements to be an involution $\tau :M\to M$ that switches the action of ${\mathsc{ℱ}}^{a}$ with that of ${\mathsc{V}}^{a}$. 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.

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