Related papers: Effective equidistribution of norm one elements in…

Let $K$ be a number field, and let $G\subset K^\times$ be a finitely generated subgroup. Fix some prime number $\ell$, and consider the set of primes $\mathfrak{p}$ of $K$ satisfying the following property: the reduction of $G$ modulo…

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism…

Let $K$ be a number field and $\mathcal{C}$ a full class of finite groups. We write $K^{\mathcal{C}}/K$ for the maximal pro-$\mathcal{C}$ Galois extension of $K$, and $G_K^{\mathcal{C}}$ for its Galois group. In this paper, we deal with the…

Let $k$ be a field, and suppose that $\Gamma$ is a smooth $k$-group that acts on a connected, reductive $k$-group $\widetilde G$. Let $G$ denote the maximal smooth, connected subgroup of the group of $\Gamma$-fixed points in $\widetilde G$.…

Let $K$ be a complete discrete valuation field with finite residue field of characteristic $p$, and let $D$ be a central division algebra over $K$ of finite index $d$. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers…

For a quartic primitive CM field $K$, we say that a rational prime $p$ is {\it evil} if at least one of the abelian varieties with CM by $K$ reduces modulo a prime ideal $\gerp| p$ to a product of supersingular elliptic curves with the…

Let R be a semi-local regular domain containing an infinite perfect field k, and let K be the field of fractions of R. Let G be a reductive semi-simple simply connected R-group scheme such that each of its R-indecomposable factors is…

Already Dedekind and Weber considered the problem of counting integral ideals of norm at most $x$ in a given number field $K$. Here we improve on the existing results in case $K/\mathbb Q$ is abelian and has degree at least four. For these…

Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences,…

Let $k_1,k_2$ be two fields of characteristic 0. Let $G_1$ be a split semisimple algebraic group over $k_1$, $G_2$ a split Kac--Moody group over $k_2$ and $\phi\colon G_1(k_1)\to G_2(k_2)$ an abstract embedding. We show that $\im \phi$ is a…

We compute the $RO(A)$-graded coefficients of $A$-equivariant complex and real topological $K$-theory for $A$ a finite elementary abelian $2$-group, together with all products, transfers, restrictions, power operations, and Adams…

We present approaches for calculating the precise orders of the algebraic $K$-groups $K_{4n-2}(\mathcal{O}_K)$ for a totally real abelian $K$. Along the way, we also establish a formula connecting the order of $K_{4n-2}(\mathcal{O}_E)$ of a…

Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the…

Let D be a division algebra with center F. A maximal subfield of D is defined to be a field K such that CD(K) = K; that is, K is its own centralizer in D. A maximal subfield K is said to be self-invariant if it normalises by itself, i.e.…

Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or…

We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy…

In this note, we show that the algebraic K-theory of generalized archimedean valuation rings occurring in Durov's compactification of the spectrum of a number ring is given by stable homotopy groups of certain classifying spaces. We also…

If a finite quasisimple group G with simple quotient S is embedded into a suitable classical group X through the smallest degree of a projective representation of S, then the normalizer of G in X is a maximal subgroup of X, up to two series…

A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action…

Let $K$ be a number field of degree $n$ over ${\mathbb Q}$. Then the 4-rank of the strict class group of $K$ is at least ${\text{rank}_2 \, } ({ E_{K}^{+} } / E_K^2) - \lfloor n /2 \rfloor$ where $E_K$ and ${ E_{K}^{+} }$ denote the units…