We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e., for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the proétale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the proétale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.

We prove that the determinant (pseudorepresentation) associated to the Hecke algebra of Katz modular forms of weight one and level prime to $p$ is unramified at $p$.

Hesselholt and Madsen (2004) define and study the (absolute, $p$-typical) de Rham–Witt complex in mixed characteristic, where $p$ is an odd prime. They give as an example an elementary algebraic description of the de Rham–Witt complex over ${\mathbb{Z}}_{\left(p\right)}$, $W_{\cdot }{\mathrm{\Omega }}_{{\mathbb{Z}}_{\left(p\right)}}^{\bullet }$. The main goal of this paper is to construct, for $k$ a perfect ring of characteristic $p>2$, a Witt complex over $A=W\left(k\right)$ with an algebraic description which is completely analogous to Hesselholt and Madsen’s description for ${\mathbb{Z}}_{\left(p\right)}$. Our Witt complex is not isomorphic to the de Rham–Witt complex; instead we prove that, in each level, the de Rham–Witt complex over $W\left(k\right)$ surjects onto our Witt complex, and that the kernel consists of all elements which are divisible by arbitrarily high powers of $p$. We deduce an explicit description of $W_n{\mathrm{\Omega }}_A^{\bullet }$ for each $n\ge 1$. We also deduce results concerning the de Rham–Witt complex over certain $p$-torsion-free perfectoid rings.

We establish Tannaka duality for noetherian algebraic stacks with affine stabilizer groups. Our main application is the existence of $\underset{¯}{Hom}$-stacks in great generality.

Let $F∕F_0$ be a quadratic extension of nonarchimedean locally compact fields of residual characteristic $p\ne 2$ and let $\sigma$ denote its nontrivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $\pi$ of ${GL}_n\left(F\right)$, with coefficients in $R$, such that ${\pi }^{\sigma }\simeq {\pi }^{\vee }$ (such a representation is said to be $\sigma$-selfdual) we associate a quadratic extension $D∕D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times }$. When $\pi$ is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for $\pi$ to be ${GL}_n\left(F_0\right)$-distinguished. When the characteristic $\ell$ of $R$ is not $2$, denoting by $\omega$ the nontrivial $R$-character of $F_0^{\times }$ trivial on $F∕F_0$-norms, we prove that any $\sigma$-selfdual supercuspidal $R$-representation is either distinguished or $\omega$-distinguished, but not both. In the modular case, that is when $\ell >0$, we give examples of $\sigma$-selfdual cuspidal nonsupercuspidal representations which are not distinguished nor $\omega$-distinguished. In the particular case where $R$ is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan, Kable and Tandon, when $p\ne 2$.

In this paper we prove a general vanishing result for Kohlhaase’s higher smooth duality functors $S^i$. If $G$ is any unramified connected reductive $p$-adic group, $K$ is a hyperspecial subgroup, and $V$ is a Serre weight, we show that $S^i\left({ind}_K^{G}V\right)=0$ for $i>dim\left(G∕B\right)$, where $B$ is a Borel subgroup and the dimension is over ${\mathbb{Q}}_p$. This is due to Kohlhaase for ${GL}_2\left({\mathbb{Q}}_p\right)$, in which case it has applications to the calculation of $S^i$ for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.