We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota–Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.

Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the pluricanonical section index${\it\delta}(V):=\text{min}\{m\mid P_{m}\geqslant 2\}$ since $1\leqslant {\it\delta}(V)\leqslant 18$ due to our previous series (I, II). Based on our further classification to 3-folds with ${\it\delta}(V)\geqslant 13$ and an intensive geometrical investigation to those with ${\it\delta}(V)\leqslant 12$, we prove that $\text{Vol}(V)\geqslant \frac{1}{1680}$ and that the pluricanonical map ${\rm\Phi}_{m}$ is birational for all $m\geqslant 61$, which greatly improves known results. An optimal birationality of ${\rm\Phi}_{m}$ for the case ${\it\delta}(V)=2$ is obtained. As an effective application, we study projective 4-folds of general type with $p_{g}\geqslant 2$ in the last section.

The goal of this paper is to obtain restrictions on the prime-to-$p$ quotient of the étale fundamental group of a smooth projective variety in characteristic $p\geqslant 0$. The results are analogues of some theorems from the study of Kähler groups. Our first main result is that such groups are indecomposable under coproduct. The second result gives a classification of the pro-$\ell$ parts of one-relator groups in this class.

Consider a vector bundle with connection on a $p$-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author’s 2010 book and on subsequent improvements by Baldassarri and by Poineau and Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point.

A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the identity of $G$ closer than $|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group $G$ is said to be Diophantine if for every $k\geqslant 1$ a random $k$-tuple in $G$ generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most $5$, or derived length at most $2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.