Related papers: Algorithmic complexity of Greenberg's conjecture

Let $p$ be a sufficiently large prime number, $r$ be any given positive integer. Suppose that $a_1,\,\dots,\,a_r$ are pairwise distinct and not zero modulo $p$. Let $N(a_1,\,\dots,\,a_r;\,p)$ denote the number of…

Let $k$ be a real abelian number field and $p$ an odd prime not dividing $[k:\mathbb{Q}]$. For a natural number $d$, let $E_d$ denote the group of units of $k$ congruent to $1$ modulo $d$, $C_d$ the subgroup of $d$-circular units of $E_d$,…

Our primary goal in this article is to study the Iwasawa theory for semi-ordinary families of automorphic forms on $\mathrm{GL}_2\times\mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$, where $K$ is an imaginary quadratic field where the prime $p$…

We prove the Goldman-Parker Conjecture: A complex hyperbolic ideal triangle group is directly embedded in PU(2,1) if and only if the product of its three standard generators is not elliptic. We also prove that such a group is indiscrete if…

Let $p$ be a prime, $k$ an algebraic closure of $\mathbb{F}_p$ and $\Gamma$ the Galois group ${\rm Gal}(k/\mathbb{F}_p)$. Let $\mathcal{C}$ be a finite category and $\mathcal{O}_{\mathcal{C}}$ the $p$-orbit category of $\mathcal{C}$ defined…

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_p$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over…

The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of algebraically trivial…

We introduce the logics GLP(\Lambda), a generalization of Japaridze's polymodal provability logic GLP(\omega) where \Lambda is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall…

We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Pad\'e approximation. In conjunction with Honda's proof of the…

Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently…

We introduce a new ideal {\mathfrak D} of the p-adic Galois group-ring associated to a real abelian field and a related ideal {\mathfrak J} for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark…

Let p be a prime and K be a number field with non-trivial p-class group Cl(p,K). A crucial step in identifying the Galois group G=G(p,K) of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G(p,2,K),…

Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\bar{\rho}_{f}$ is irreducible. Under some additional hypotheses we prove that…

Let $p$ be an odd prime. We prove the cyclotomic Iwasawa Main Conjecture of K.Kato for the motive attached to an eigencuspform $f\in S_{k}(\Gamma_{0}(N))$ with arbitrary reduction type at $p$ under mild assumptions on the residual Galois…

Let $\Theta$ be a finite alphabet. We consider a bundle of measure preserving transformations $(T_{\theta})_{\theta \in \Theta}$ acting on a probability space $(X,\mu)$, which are chosen randomly according to an ergodic stochastic process…

For every natural number k we prove a decomposition theorem for bounded measurable functions on compact abelian groups into a structured part, a quasi random part and a small error term. In this theorem quasi randomness is measured with the…

We study a stochastic complex Ginzburg--Landau (CGL) equation driven by a smooth noise in space and we establish exponential convergence of the Markovian transition semi-group toward a unique invariant probability measure. Since Doob…

For a given graph $G$ of minimum degree at least $k$, let $G_p$ denote the random spanning subgraph of $G$ obtained by retaining each edge independently with probability $p=p(k)$. We prove that if $p \ge \frac{\log k + \log \log k +…

Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the…

Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a…