For a compact $3$-manifold $M$ with arbitrary (possibly empty) boundary, we give a parameterization of the set of conjugacy classes of boundary-unipotent representations of ${\pi }_1\left(M\right)$ into $SL(n,\mathbb{C})$. Our parameterization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmüller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann’s extended Bloch group $\hat{\mathcal{B}}\left(\mathbb{C}\right)$, and we use this to obtain an efficient formula for the Cheeger–Chern–Simons invariant, and, in particular, for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic $3$-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.

Let $X$ be a smooth proper variety of even dimension $d$ over a finite field. We establish a restriction on the value at $(-1)$ of the characteristic polynomial of the Frobenius on the middle-dimensional étale cohomology of $X$ with coefficients in ${\mathbb{Q}}_l(d/2)$.

We prove that the field ${\mathbf{Q}}_p$ and totally imaginary number fields satisfy the $C_1^1$ property conjectured by Kato and Kuzumaki in 1986. In other words, if $k$ denotes one of these fields and $f\in k[x_0,\dots ,x_n]$ is a homogeneous polynomial of degree $d\le n$, every element of $k$ may be written as a product of norms from finite extensions of $k$ in which $f$ possesses a nontrivial zero. We also establish Ax’s conjecture about perfect pseudoalgebraically closed fields for fields whose absolute Galois group is a pro-$p$-group.

Nous montrons que le corps ${\mathbf{Q}}_p$ et les corps de nombres totalement imaginaires vérifient la propriété $C_1^1$ conjecturée par Kato et Kuzumaki en 1986. Autrement dit, si $k$ est l’un de ces corps et $f\in k[x_0,\dots ,x_n]$ est un polynôme homogène de degré $d\le n$, tout élément de $k$ s’écrit comme produit de normes depuis des extensions finies de $k$ dans lesquelles $f$ admet un zéro non trivial. Nous établissons aussi la conjecture d’Ax sur les corps pseudo-algébriquement clos parfaits pour les corps dont le groupe de Galois absolu est un pro-$p$-groupe.

We give a description of the boundary of a complex of free factors that is analogous to E. Klarreich’s description of the boundary of a curve complex. The argument uses the geometry of folding paths developed by Bestvina and Feighn and the structure theory of trees on the boundary of Outer space developed recently by Coulbois, Hilion, Lustig, and Reynolds.