Related papers: Some applications of modular units

In contrast to that a weak value of an observable is usually divided into real and imaginary parts, here we show that separation into modulus and argument is important for modular values. We first show that modular values are expressed by…

Let $V$ be a strongly regular vertex operator algebra. For a state $h \in V_1$ satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr$_Mq^{L(0)-c/24}\zeta^{h(0)} ($M$ a $V$-module) is a…

We consider the moduli space $A_{pol}(n)$ of (non-principally) polarised abelian varieties of genus $g\geq3$ with coprime polarisation and full level-$n$ structure. Based upon the analysis of the Tits building in math/0405321, we give an…

We present the first example of the Selberg type zeta function for noncompact higher rank locally symmetric spaces. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real…

We prove that there is a continuous non-negative function $g$ on the unit sphere in $\cd$, $d \geq 2$, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded…

We study operator algebraic and function theoretic aspects of algebras of bounded nc functions on subvarieties of the nc domain determined by all levels of the unit ball of an operator space (nc operator balls). Our main result is the…

Let $\cl{M}$ be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(\Omega)$, where $\Omega \subseteq \bb{C}^m$ is a bounded domain. Let $\cl{M}_0\subseteq \cl{M}$ be the submodule of functions vanishing…

The paper proposes a vector generalization of the basic concepts of the theory of complex variable: the concept of modulus and argument of complex number. The author introduces some generalizations of the notion of holomorphic functions and…

We prove that only finitely many $j$-invariants of elliptic curves with complex multiplication are algebraic units. A rephrased and generalized version of this result resembles Siegel's Theorem on integral points of algebraic curves.

A solution to the part and whole problem is presented in this paper by using a complex mathematical representation that permits to define the Holon concept as a Basic Unit System that remains itself in spite of complex operations such as…

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of…

The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $E_{8}$ or its highest weight is minuscule. In this paper, we prove…

False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among…

Let $O_L$ be the ring of integers of a number field $L$. Write $q = e^{2 \pi i z}$, and suppose that $$f(z) = \sum_{n \gg - \infty}^{\infty} a_f(n) q^n \in M_{k}^{!}(\operatorname{SL}_2(\mathbb{Z})) \cap O_L[[q]]$$ is a weakly holomorphic…

Following Zagier, this work studies the rationality and divisibility of Fourier coefficients of meromorphic Hilbert modular forms associated with real quadratic fields, using theta lifts and weak Maass forms. We establish conditions where…

Let $T$ be a Noetherian ring and $f$ a nonzerodivisor on $T$. We study concrete necessary and sufficient conditions for a module over $R=T/(f)$ to be weakly liftable to $T$, in the sense of Auslander, Ding and Solberg. We focus on cyclic…

The fractional polylogarithms, depending on a complex parameter $\a$, are defined by a series which is analytic inside the unit disk. After an elementary conversion of the series into an integral presentation, we show that the fractional…

We find some modularity criterion for a product of Klein forms of the congruence subgroup $\Gamma_1(N)$ and, as its application, construct a basis of the space of modular forms for $\Gamma_1(13)$ of weight $2$. In the process we face with…

We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp…

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized…