A twisted cocycle taking values on a Lie group G is a cocycle that is twisted by an automorphism of G in each step. In the case where G = GL(d, ℝ), we prove that if two Hölder continuous twisted cocycles satisfying the so-called fiber-bunching condition have the same periodic data then they are cohomologous.

We correct the proof of the main $\ell$ -independence result of the above-mentioned paper by showing that for any smooth and proper variety over an equicharacteristic local field, there exists a globally defined such variety with the same ( $p$ -adic and $\ell$ -adic) cohomology.

We give a geometric interpretation of sheaf cohomology for higher degrees $n\geq 1$ in terms of torsors on the member of degree $d=n-1$ in hypercoverings of type $r=n-2$ , endowed with an additional datum, the so-called rigidification. This generalizes the fact that cohomology in degree one is the group of isomorphism classes of torsors, where the rigidification becomes vacuous, and that cohomology in degree two can be expressed in terms of bundle gerbes, where the rigidification becomes an associativity constraint.

We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including $\overline{\mathbb{Q}}$ and $\overline{\mathbb{F}_{p}}$ , we arrive at a complete description of the tensor triangular spectrum and a classification of the thick tensor ideals.

Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated with the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated with the Weyl chambers of type Cn and Dn, completing the computation for all classical types.

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct $p$-adic $L$-functions for non-critical slope rational modular forms, the theory has been extended to construct $p$-adic $L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the $L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the $p$-adic $L$-function of the eigenform to be this distribution.

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative property (T) for a subgroupoid $H\subset G$ in terms of the inclusions $L(H)\subset L(G)$.

In this article we study various forms of $\ell$ -independence (including the case $\ell =p$ ) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of $\ell$ -independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain ‘spreading out’ result we are able to deduce a much weaker form of $\ell$ -independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce $\ell$ -independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on $\ell$ -independence for smooth and proper varieties over finite fields. As another consequence of this ‘spreading out’ result we are able to deduce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic $p$ , we show a similar weak version of $\ell$ -independence for the unipotent fundamental group of a semistable curve in mixed characteristic.

We determine the Galois representations inside the $\ell$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at $p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.

We prove that, under rather general conditions, the 1-cohomology of a von Neumann algebra $M$ with values in a Banach $M$ -bimodule satisfying a combination of smoothness and operatorial conditions vanishes. For instance, we show that, if $M$ acts normally on a Hilbert space ${\mathcal{H}}$ and ${\mathcal{B}}_{0}\subset {\mathcal{B}}({\mathcal{H}})$ is a norm closed $M$ -bimodule such that any $T\in {\mathcal{B}}_{0}$ is smooth (i.e., the left and right multiplications of $T$ by $x\in M$ are continuous from the unit ball of $M$ with the $s^{\ast }$ -topology to ${\mathcal{B}}_{0}$ with its norm), then any derivation of $M$ into ${\mathcal{B}}_{0}$ is inner. The compact operators are smooth over any $M\subset {\mathcal{B}}({\mathcal{H}})$ , but there is a large variety of non-compact smooth elements as well.

Let $G$ be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety $G/B$ . The main theorem of this note gives an eõcient presentation of the equivariant cohomology ring $H_{S}^{*}$ (Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal $J$ generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group $S\,\cong \,{{\mathbb{C}}^{*}}$ is a certain subgroup of a maximal torus $T$ of $G$ . Our description of the ideal $J$ uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa–Harada–Masuda, which was only for Lie type $A$ .

We apply the endoscopic classification of automorphic forms on $U(3)$ to study the growth of the first Betti number of congruence covers of a Picard modular surface. As a consequence, we establish a case of a conjecture of Sarnak and Xue on cohomology growth.

In this work, we s uggest a defnition for the category of mixed motives generated by the motive h1 (E) for E an elliptic curve without complex multiplication. We then compute the cohomology of this category. Modulo a strengthening of the Beilinson-Soulé conjecture, we show that the cohomology of our category agrees with the expected motivic cohomology groups. Finally for each pure motive (Symnh1 (E)) (–1) we construct families of nontrivial motives whose highest associated weight graded piece is (Symnh1 (E)) (–1).

In [TVa], Bertrand Toën and Michel Vaquié defined a scheme theory for a closed monoidal category ( ⊗1). In this article, we define a notion of smoothness in this relative (and not necessarily additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists in replacing homological finiteness conditions by homotopical ones, using the Dold-Kan correspondence. To do this, we provide the category s of simplicial objects in a monoidal category and all the categories sA-mod, sA-alg (A ∈ sComm()) with compatible model structures using the work of Rezk [R]. We then give a general notion of smoothness in sComm(). We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and is stable under composition and homotopy pushouts. Finally we provide some examples of smooth morphisms, in particular in ℕ-alg and Comm(Set).

We compute the rings ${H}^{\ast } (N; { \mathbb{F} }_{2} )$ for $N$ a closed $ \mathbb{S} {\mathrm{ol} }^{3} $-manifold, and then determine the Borsuk–Ulam indices $\text{BU} (N, \phi )$ with $\phi \not = 0$ in ${H}^{1} (N; { \mathbb{F} }_{2} )$.

In this paper we prove that surfaces of general type with irregularity $q\geq 3$ are rationally cohomologically rigidified, and so are minimal surfaces $S$ with $q(S)= 2$ unless ${ K}_{S}^{2} = 8\chi ({ \mathcal{O} }_{S} )$. Here a surface $S$ is said to be rationally cohomologically rigidified if its automorphism group $\mathrm{Aut} (S)$ acts faithfully on the cohomology ring ${H}^{\ast } (S, \mathbb{Q} )$. As examples we give a complete classification of surfaces isogenous to a product with $q(S)= 2$ that are not rationally cohomologically rigidified.

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? In this paper we provide, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens–Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.