Related papers: Effective bounds on differences of singular moduli…

We show that the Julia set of a non-elementary rational semigroup G is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of G. This also proves that the limit set of a non-elementary Moebius group…

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such…

We develop $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of cubic Hecke $L$-functions of prime moduli over the Eisenstein field using multiple Dirichlet series under the…

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_\infty$-invariant and concentrated on a single…

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for…

For a rational map $\phi: X \to G$ from a normal algebraic variety $X$ to a commutative algebraic group $G$, we define the modulus of $\phi$ as an effective divisor on $X$. We study the properties of the modulus. This work generalizes the…

A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural…

We prove a fixed point theorem for the action of certain local monodromy groups on \'etale covers and use it to deduce lower bounds in essential dimension. In particular, we give more geometric proofs of many (but not all) of the results of…

Based on results obtained in a companion paper [MSRI preprint 1997-002], we construct groups of special $S$--units for function fields of characteristic $p>0$, and show that they satisfy Gras--type Conjectures. We use these results in order…

We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is stratifed according to genus, and it carries a metric and a measure that…

After Voronin proved the universality theorem of the Riemann zeta function in the 1970s, universality theorems have been proposed for various zeta and L-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universality theorem of…

We continue the investigation into the computational status of the existence of moduli of regularity (and their use for rates of convergence) in the sense of Kohlenbach, Lopez and Nicolae (2019), carried out w.r.t. classical reverse…

Given a unital ring $R$ and a two-sided ideal $I$ of $R$, we consider the question of determining when a unit of $R/I$ can be lifted to a unit of $R$. For the wide class of separative exchange ideals $I$, we show that the only obstruction…

We describe the support of $F$-finite $F$-modules over polynomial rings $R$ of prime characteristic. Our description yields an algorithm to compute the support of such modules; the complexity of our algorithm is also analyzed. To the best…

We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible…

Let $G$ be a finite group and $\rho:G \to \GL(V)$ a finite dimensional representation of $G$. We say that $\rho$ is unisingular if $\det(1-\rho(g)) = 0$ for all $g \in G$. Building on previous work in \cite{cullinan}, we consider the…

Let $R$ be an affine algebra over an algebraically closed field of characteristic $0$ with dim$(R)=n$. Let $P$ be a projective $A=R[T_1,\cdots,T_k]$-module of rank $n$ with determinant $L$. Suppose $I$ is an ideal of $A$ of height $n$ such…

Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. Let $\mathcal{S}(M)$ be the set of all submodules of $M$, and $\phi:\mathcal{S}(M)\to \mathcal{S}(M)\cup \{\emptyset\}$ be a function. We say that a…

Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse…

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the…