We define an “antiholomorphic involution” of a module
over the
Dieudonné ring
of
a finite field
with
elements to be an
involution
that
switches the action of
with that of
.
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 , in which case
the contribution at
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.
