相关论文: On an estimate on G\"otzky's domain

We study the action of Bianchi groups on the hyperbolic $3$-space $\mathbb{H}^3$. Given the standard fundamental domain for this action and any point in $\mathbb{H}^3,$ we show that there exists an element in the group which sends the given…

We continue our study of integral points on moduli schemes by combining the method of Faltings (Arakelov, Parsin, Szpiro) with modularity results and Masser-W\"ustholz isogeny estimates. In this work we explicitly bound the height and the…

We prove a linear upper bound for the number of singular points on the boundary of a quadrature domain, improving a previously known quadratic bound due to Gustafsson \cite{Gus88}. This linear upper bound on the number of boundary double…

We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a…

Let $\mathbf{G}$ be a reductive group defined over $\mathbb{Q}$ and let $\mathfrak{S}$ be a Siegel set in $\mathbf{G}(\mathbb{R})$. The Siegel property tells us that there are only finitely many $\gamma \in \mathbf{G}(\mathbb{Q})$ of…

The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\widehat{h}(F)$ bounded below by a constant. It was first proved by Breuillard and another proof was…

This study utilizes Orlicz functions to provide refined lower and upper bounds on the q-numerical radius of an operator acting on a Hilbert space. Additionally, the concept of q-sectorial matrices is introduced and further bounds for the…

The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field $\mathbb{Q}(\sqrt{-5460})$ has finite or infinite $2$-class tower. This is a critical case…

Inspired by the Odlyzko root discriminant and Golod--Shafarevich $p$-group bounds, Martinet (1978) asked whether an imaginary quadratic number field $K/\mathbb{Q}$ must always have an infinite Hilbert $2$-class field tower when the class…

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a…

We generalize the absolute logarithmic Weil height from elements of the multiplicative group of algebraic numbers modulo torsion, to finitely generated subgoups. The height of a finitely generated subgroup is shown to equal the volume of a…

The purpose of this paper is to provide a detailed description of the spaces that can be specified as $L^2$ domains for the operators of a first order elliptic complex on a compact manifold with conical singularities. This entails an…

The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an…

It is well known that, in the plane, the boundary of any quadrature domain (in the classical sense) coincides with the zero set of a polynomial. We show, by explicitly constructing some four-dimensional examples, that this is not always the…

In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $\Gamma$ be a congruence subgroup of $\SL_2(\Om_K)$…

We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form…

The Hot Spots constant for bounded smooth domains was recently introduced by Steinerberger (2021) as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We generalize…

For integers $a\neq0$, $k$, and $n\geq3$, we consider the Markoff-Hurwitz equation given by $x_1^1+\cdots+x_n^2-ax_1\cdots x_n=k$. By defining graphs associated with a height function and by using their properties, we find an exact…

We use the method of Faltings (Arakelov, Par\v{s}in, Szpiro) in order to explicitly study integral points on a class of varieties over $\mathbb Z$ called Hilbert moduli schemes. For instance, integral models of Hilbert modular varieties are…

We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and H\"older-Zygmund spaces for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$, extending the…