Fix a number field $F\subset ℂ$, an abelian variety $A∕F$ and let $G_A$ be the Mumford–Tate group of $A_{∕ℂ}$. After replacing $F$ by finite extension one can assume that, for every prime number $\ell$, the action of the absolute Galois group ${\Gamma }_F=Gal\left(\bar{F}∕F\right)$ on the étale cohomology group $H_{ét}^1\left(A_{\bar{F}},{\mathbb{Q}}_{\ell }\right)$ factors through a morphism ${\rho }_{\ell }:{\Gamma }_F\to G_A\left({\mathbb{Q}}_{\ell }\right)$. Let $v$ be a valuation of $F$ and write ${\Gamma }_{F_v}$ for the absolute Galois group of the completion $F_v$. For every $\ell$ with $v\left(\ell \right)=0$, the restriction of ${\rho }_{\ell }$ to ${\Gamma }_{F_v}$ defines a representation ${}^{\prime }{W}_{F_v}\to G_{A∕{\mathbb{Q}}_l}$ of the Weil–Deligne group.

It is conjectured that, for every $\ell$, this representation of ${}^{\prime }{W}_{F_v}$ is defined over $\mathbb{Q}$ as a representation with values in $G_A$ and that the system above, for variable $\ell$, forms a compatible system of representations of ${}^{\prime }{W}_{F_v}$ with values in $G_A$. A somewhat weaker version of this conjecture is proved for the valuations of $F$, where $A$ has semistable reduction and for which ${\rho }_{\ell }\left({Fr}_v\right)$ is neat.

This paper studies the Fourier–Jacobi expansions of Eisenstein series on $U\left(3,1\right)$. I relate the Fourier–Jacobi coefficients of the Eisenstein series with special values of $L$-functions. This relationship can be applied to verify the existence of certain Eisenstein series on $U\left(3,1\right)$ that do not vanish modulo $p$. This is a crucial step towards one divisibility of the main conjecture for ${GL}_2\times K^{\times }$ using the method of Eisenstein congruences.

A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. We describe the structure of the boundary of the coamoeba of a variety, which we relate to its logarithmic limit set. Detailed examples of lines in three-dimensional space illustrate and motivate these results.

We show that the compatibility of the relative canonical sheaf with base change fails generally in families of normal varieties. Furthermore, it always fails if the general fiber of a family of pure dimension $n$ is Cohen–Macaulay and the special fiber contains a strictly $S_{n-1}$ point. In particular, in moduli spaces with functorial relative canonical sheaves Cohen–Macaulay schemes can not degenerate to $S_{n-1}$ schemes. Another, less immediate consequence is that the canonical sheaf of an $S_{n-1}$, $G_2$ scheme of pure dimension $n$ is not $S_3$.

We provide sufficient conditions for the line bundle locus in a family of compact moduli spaces of pure sheaves to be isomorphic to the Néron model. The result applies to moduli spaces constructed by Eduardo Esteves and Carlos Simpson, extending results of Busonero, Caporaso, Melo, Oda, Seshadri, and Viviani.

The arithmetic motivic Poincaré series of a variety $V$ defined over a field of characteristic zero is an invariant of singularities that was introduced by Denef and Loeser by analogy with the Serre–Oesterlé series in arithmetic geometry. They proved that this motivic series has a rational form that specializes to the Serre–Oesterlé series when $V$ is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper, we study this motivic series when $V$ is an affine toric variety. We obtain a formula for the rational form of this series in terms of the Newton polyhedra of the ideals of sums of combinations associated to the minimal system of generators of the semigroup of the toric variety. In particular, we explicitly deduce a finite set of candidate poles for this invariant.

Given a central simple algebra $\mathfrak{g}$ and a Galois extension of base rings $S∕R$, we show that the maximal ideals of twisted $S∕R$-forms of the algebra of currents $\mathfrak{g}\left(R\right)$ are in natural bijection with the maximal ideals of $R$. When $\mathfrak{g}$ is a Lie algebra, we use this to give a complete classification of the finite-dimensional simple modules over twisted forms of $\mathfrak{g}\left(R\right)$.

We give explicit descriptions of the higher Chow groups of toric bundles and flag bundles over schemes. We derive several consequences of these descriptions for the equivariant and ordinary higher Chow groups of schemes with group action.

We prove a decomposition theorem for the equivariant higher Chow groups of a smooth scheme with action of a diagonalizable group. This theorem is applied to compute the equivariant and ordinary higher Chow groups of smooth toric varieties. The results of this paper play fundamental roles in the proof of the Riemann–Roch theorems for equivariant higher $K$-theory.