Related papers: A Thin Fundamental Set for SL(2, Z)

We extend the framework of modular invariant supersymmetric theories to encompass invariance under more general discrete groups $\Gamma$, that allow the presence of several moduli and make connection with the theory of automorphic forms.…

We formulate integral Iwasawa main conjectures for suitable twists of a newform $f$ that is non-ordinary at $p$, over the cyclotomic $\mathbb{Z}_p$-extension, the anticyclotomic $\mathbb{Z}_p$-extensions (in both the definite and the…

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $\delta(G)$ and $\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in…

Wang and Sun proved a certain summatory formula involving derangements and primitive roots of the unit. We study such a formula but for the particular case of the set of affine derangements in $\overrightarrow{GL}(\mathbb{Z}/2k\mathbb{Z})$…

The Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and only if $G$ contains an element of order $pq$. This graph has been the subject of much…

We present an analogue of Greenberg-Vatsal's and Emerton-Pollack-Weston's results on congruences of $p$-adic $L$-functions for $p$-non-ordinary cuspidal eigenforms $f$ and $g$ of equal weight that are $p$-congruent. In particular, we prove…

Let $\ell$ be a prime number, $k$ a positive integer and consider the group $\Gamma_{\ell^k} :=\langle a,b\ \vert\ a^{\ell^k(\ell^k-1)}ba^{-\ell^k}b^{-2}\rangle$. We prove that $\Gamma_{\ell^k}$ is not $\mathrm{SL}_2$-weakly integral with…

Let $G$ be a finite group. We let $\f{m}(G)$ and $\sig(G)$ denote the number of maximal subgroups of $G$ and the least positive integer $n$ such that $G$ is written as the union of $n$ proper subgroups, respectively. In this paper we…

We prove the following result due to Hamidoune using an analytic approach. Suppose that A is a subset of a finite group G with |AA^{-1}| \leq (2-\varepsilon)|A|. Then there is a subgroup H of G and a set X of size O_\varepsilon(1) such that…

Let Gamma be a Q-polynomial distance-regular graph with vertex set X, diameter D geq 3 and adjacency matrix A. Fix x in X and let A*=A*(x) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T=T(x) is the…

We initiate a study of the spectral theory of the locally symmetric space $X=\Gamma\backslash G/K$, where $G=SO(3,Complex)$, $\Gamma=SO(3,Z[i])$, $K=SO{3}$. We write down explicit equations defining a fundamental domain for the action of…

We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form $\Gamma\times\Gamma'$ for $\Gamma,\Gamma'\in {\mathbf\Sigma^0_1 , \mathbf\Pi^0_1 , \mathbf\Pi^0_2 }$.

We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in…

We derive the equation of matter density perturbations on sub-horizon scales around a flat Friedmann-Lema\^\i tre-Robertson-Walker background for the general Lagrangian density $f(R,\GB)$ that is a function of a Ricci scalar $R$ and a…

Let $n \ge 2$, $p \in (1, \, +\infty)$ be given and let $\Sigma$ be a $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$. Denote by $A$ its second fundamental form, and by $\mathring{A}$ the tensor $A - \frac{1}{n} A^i_i g$ where $g…

We prove part of a higher rank analogue of the Mazur-Gouvea Conjecture. More precisely, let $\tilde{\bf G}$ be a connected, reductive ${\Bbb Q}$-split group and let $\Gamma$ be an arithmetic subgroup of $\tilde{\bf G}$. We show that the…

Let $E \subset B(1) \subset \mathbb R^{2}$ be an $\mathcal{H}^{1}$ measurable set with $\mathcal{H}^{1}(E) < \infty$, and let $L \subset \mathbb R^{2}$ be a line segment with $\mathcal{H}^{1}(L) = \mathcal{H}^{1}(E)$. It is not hard to see…

Let K be a number field. For any system of semisimple mod l Galois representations {\phi_l:Gal_K->GL_N(F_l)} arising from \'etale cohomology, there exists a finite normal extension L of K such that if we denote \phi_l(Gal_K) and…

Suppose that $\Gamma$ is a conformal loop ensemble (CLE$_\kappa$) with simple loops ($\kappa \in (8/3,4)$) in a simply connected domain $D \subseteq {\mathbf C}$ whose boundary is itself a type of CLE$_\kappa$ loop. Let $\Upsilon$ be the…

We call an element of a finite general linear group $ \textrm{GL}(d,q) $ \emph{fat} if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than $d/2$. Fatness of an element can be decided efficiently in practice by…