We describe the algebraic de Rham realization of the elliptic polylogarithm for arbitrary families of elliptic curves in terms of the Poincaré bundle. Our work builds on previous work of Scheider and generalizes results of Bannai, Kobayashi and Tsuji, and Scheider. As an application, we compute the de Rham–Eisenstein classes explicitly in terms of certain algebraic Eisenstein series.

Let $\pi :Y\to X$ be a branched $\mathbf{Z}∕p\mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $\pi$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann–Hurwitz (respectively Deuring–Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the $a$-number of $Y$. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.

We consider the family of irreducible crystalline representations of dimension $2$ of $Gal\left({\bar{\mathbb{Q}}}_p∕{\mathbb{Q}}_p\right)$ given by the $V_{k,a_p}$ for a fixed weight $k\ge 2$. We study the locus of the parameter $a_p$ where these representations have a given reduction modulo $p$. We give qualitative results on this locus and show that for a fixed $p$ and $k$ it can be computed by determining the reduction modulo $p$ of $V_{k,a_p}$ for a finite number of values of the parameter $a_p$. We also generalize these results to other Galois types.

Let $K$ be a number field or a $p$-adic field and $F$ the function field of a curve over $K$. Let $\ell$ be a prime. Suppose that $K$ contains a primitive $\ell$-th root of unity. If $\ell =2$ and $K$ is a number field, then assume that $K$ is totally imaginary. In this article we show that every element in $H^3\left(F,{\mu }_{\ell }^{\otimes 3}\right)$ is a symbol. This leads to the finite generation of the Chow group of zero-cycles on a quadric fibration of a curve over a totally imaginary number field.

We introduce a certain birational invariant of a polarized algebraic variety and use that to obtain upper bounds for the counting functions of rational points on algebraic varieties. Using our theorem, we obtain new upper bounds of Manin type for $28$ deformation types of smooth Fano $3$-folds of Picard rank $\ge 2$ following the Mori–Mukai classification. We also find new upper bounds for polarized K3 surfaces $S$ of Picard rank $1$ using Bayer and Macrì’s result on the nef cone of the Hilbert scheme of two points on $S$.

Baker’s method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we generalize results of Levin regarding Baker’s method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge’s method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in $p$-adic logarithms. We then use these ideas to improve known estimates on solutions of $S$-unit equations. Finally, we explain how a finer analysis and formalism can improve upon the conditions given, and give some applications to the Siegel modular variety $A_2\left(2\right)$.

We study (smooth, complex) Fano 4-folds $X$ having a rational contraction of fiber type, that is, a rational map $X--\to Y$ that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a nonzero, nonbig movable divisor on $X$. Our main result is that if $Y$ is not ${\mathbb{P}}^1$ or ${\mathbb{P}}^2$, then the Picard number ${\rho }_X$ of $X$ is at most 18, with equality only if $X$ is a product of surfaces. We also show that if a Fano 4-fold $X$ has a dominant rational map $X--\to Z$, regular and proper on an open subset of $X$, with $dim\left(Z\right)=3$, then either $X$ is a product of surfaces, or ${\rho }_X$ is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.