Let $K$ be a finite field extension of ${\mathbb{Q}}_p$ and let ${\mathcal{𝒢}}_K$ be its absolute Galois group. We construct the universal family of filtered $\left(\varphi ,N\right)$-modules, or (more generally) the universal family of $\left(\varphi ,N\right)$-modules with a Hodge–Pink lattice, and study its geometric properties. Building on this, we construct the universal family of semistable ${\mathcal{𝒢}}_K$-representations in ${\mathbb{Q}}_p$-algebras. All these universal families are parametrized by moduli spaces which are Artin stacks in schemes or in adic spaces locally of finite type over ${\mathbb{Q}}_p$ in the sense of Huber. This has conjectural applications to the $p$-adic local Langlands program.

Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We describe several mild geometric conditions ensuring that the group $A\left(K^{perf}\right)$ is finitely generated and that the $p$-primary torsion subgroup of $A\left(K^{sep}\right)$ is finite. This gives partial answers to questions of Scanlon, Ghioca and Moosa, and Poonen and Voloch. We also describe a simple theory (used to prove our results) relating the Harder–Narasimhan filtration of vector bundles to the structure of finite flat group schemes of height one over projective curves over perfect fields. Finally, we use our results to give a complete proof of a conjecture of Esnault and Langer on Verschiebung divisibility of points in abelian varieties over function fields.

Over the past fifteen years or so, Minhyong Kim has developed a framework for making effective use of the fundamental group to bound (or even compute) integral points on hyperbolic curves. This is the third installment in a series whose goal is to realize the potential effectivity of Kim’s approach in the case of the thrice punctured line. As envisioned by Dan-Coehn and Wewers (2016), we construct an algorithm whose output upon halting is provably the set of integral points, and whose halting would follow from certain natural conjectures. Our results go a long way towards achieving our goals over the rationals, while broaching the topic of higher number fields.

We study the asymptotic distribution of CM points on the moduli space of elliptic curves over ${ℂ}_p$, as the discriminant of the underlying endomorphism ring varies. In contrast with the complex case, we show that there is no uniform distribution. In this paper we characterize all the sequences of discriminants for which the corresponding CM points converge towards the Gauss point of the Berkovich affine line. We also give an analogous characterization for Hecke orbits. In the companion paper we characterize all the remaining limit measures of CM points and Hecke orbits.

We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field ${\mathbb{𝔽}}_q$, for almost all couples of arguments in ${\mathbb{𝔽}}_q^{\times }$, as well as lower bounds on differences. Using similar ideas, we then study the biases in the distribution of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick and Waxman, and Keating and Rudnick, building on cohomological interpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenvalues of $\ell$-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties.