Abstract: Lazard showed that the continuous group cohomology of many p-adic Lie groups satisfies Poincaré duality.  Like for non-orientable manifolds, there is a twist in the Poincaré duality; but, crucially, Lazard was able to concretely identify this twist.  In particular, he saw that it only depends on the adjoint representation of G, a kind of ``linearization" of the problem which permits efficient calculation.  We will discuss what happens when one passes to spectrum coefficients.  There is again Poincaré duality up to a twist, but identifying the twist in terms of the adjoint representation is now more subtle and difficult.  The proof that this can be done involves two main ingredients: one, a sheaf-theoretic formalism for discussing Poincaré duality ``in families"; and two, a novel p-adic co-specialization map in this context.
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