Related papers: Annihilating class groups in $p$-elementary extens…

We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela).…

We prove several basic extension theorems for reductive group schemes. We also prove that each Lie algebra with a perfect Killing form over a commutative $\dbZ$-algebra, is the Lie algebra of an adjoint group scheme.

We deduce the relative version of the equivalences relating the relative Local Global Principle and the Normality of the relative Elementary subgroups of the traditional classical groups, viz. general linear, symplectic and orthogonal…

Slattery has generalized Brauer's theory of p-blocks of finite groups to pi-blocks of pi-separable groups where pi is a set of primes. In this setting we show that the order of a defect group of a pi-block B is bounded in terms of the…

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-modules. Examples of this…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…

We use our extension of the Noether-Lefschetz theorem to describe generators of the class groups at the local rings of singularities of very general hypersurfaces containing a fixed base locus. We give several applications, including (1)…

In the present work we give the construction of the genus field and the extended genus field of an elementary abelian $l$-extension of a field of rational functions, where $l$ is a prime number. In the Kummer case, if $K$ is contained in a…

Let $B \subseteq A$ be an extension of finite dimensional algebras. We provide a sufficient condition for the existence of triangle equivalences of singularity categories (resp. Gorenstein defect categories) between $A$ and $B$. This result…

This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If $RG$ is a group ring, where $R$ is commutative and $S$ is a set of generators of $G$ then necessary and sufficient…

In this paper we study elementary extensions of differential fields in prime characteristic. In particular, we show that, in contrast to Liouville's result in characteristic zero, all elements of an elementary extension admit an…

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $p$ be an odd prime and $r>1$ be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin…

Given a random real quadratic field from $\{ \mathbb{Q}(\sqrt{p}\,) ~|~ p \text{ primes} \}$, the conjectural probability $\mathbb{P}(h=q)$ that it has class number $q$ is given for all positive odd integers $q$. Some related conjectures of…

There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochischild-Serre spectral sequence of an extension of an integral lattice L by a group G. These characteristic classes exist in…

We study the families of measures on Carnot groups that have vanishing $p$-module, which we call $p$-exceptional families. We found necessary and sufficient condition for the family of intrinsic Lipschitz surfaces passing through a common…

We present the first algorithm for computing class groups and unit groups of arbitrary number fields that provably runs in probabilistic subexponential time, assuming the Extended Riemann Hypothesis (ERH). Previous subexponential algorithms…

Ideas and techniques from Khare's and Wintenberger's article on the proof of Serre's conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL_2(F_{l^r}) (for r running) occur as Galois…

In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi(m)/(k+1)$ residue classes in the…

The P\'{o}lya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large…

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose…