Let $X/S$ be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes $H/S$ of $X/S$ with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in $X/S$ containing a given closed subscheme $C$ and intersecting properly a closed set $F$.

Assume now that the base $S$ is the spectrum of a ring $R$ such that for any finite morphism $Z\to S$, $Pic\left(Z\right)$ is a torsion group. This condition is satisfied if $R$ is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal $1$-cycles on a regular scheme $X$ quasi-projective and flat over $S$. We also show the existence of a finite surjective $S$-morphism to ${\mathbb{P}}_S^d$ for any scheme $X$ projective over $S$ when $X/S$ has all its fibers of a fixed dimension $d$.

Let $G$ be a split connected reductive algebraic group over ${\mathbb{Q}}_p$ such that both $G$ and its dual group $\hat{G}$ have connected centers. Motivated by a hypothetical $p$-adic Langlands correspondence for $G\left({\mathbb{Q}}_p\right)$, we associate to an $n$-dimensional ordinary (i.e., Borel-valued) representation $\rho :Gal({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)\to \hat{G}\left(E\right)$ a unitary Banach space representation $\Pi (\rho {)}^{\mathrm{ord}}$ of $G\left({\mathbb{Q}}_p\right)$ over $E$ that is built out of principal series representations. (Here, $E$ is a finite extension of ${\mathbb{Q}}_p$.) Our construction is inspired by the “ordinary part” of the tensor product of all fundamental algebraic representations of $G$. There is an analogous construction over a finite extension of ${\mathbb{F}}_p$. When $G=G{L}_n$, we show under suitable hypotheses that $\Pi (\rho {)}^{\mathrm{ord}}$ occurs in the $\rho$-part of the cohomology of a compact unitary group.

We prove that the Möbius function is linearly disjoint from an analytic skew product on the $2$-torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of the Möbius function from various distal homogeneous flows.

We prove weak approximation for smooth cubic hypersurfaces of dimension at least $2$ defined over the function field of a complex curve.

We discuss rescaling limits for sequences of complex rational maps in one variable which approach infinity in parameter space. It is shown that any given sequence of maps of degree $d\ge 2$ has at most $2d-2$ dynamically distinct rescaling limits which are not postcritically finite. For quadratic rational maps, a complete description of the possible rescaling limits is given. These results are obtained by employing tools from nonarchimedean dynamics.