We consider modules generated by torsion elements of a fixed normalized rank one Drinfeld module of generic characteristic. The behavior of these modules seems to be governed by a vector space whose dimension is bounded by the genus of the ground field.

A Delone set in $\mathbb{R}^n$ is a set such that (a) the distance between any two of its points is uniformly bounded below by a strictly positive constant and such that (b) the distance from any point in $\mathbb{R}^n$ to the set is uniformly bounded above.Delone sets are thus sets of points enjoying nice spacing properties, and they therefore appear naturally in mathematical models for quasicrystals. Define a spiral set in $\mathbb{R}^n$ as a set of points of the form $\{\sqrt[n]{k}\cdot\bm{u}_k\}_{k\ge 1}$, where $(\bm{u}_k)_{k\ge 1}$ is a sequence in the unit sphere $\mathbb{S}^{n-1}$. In the planar case $n=2$, spiral sets serve as natural theoretical models in phyllotaxis (the study of configurations of leaves on a plant stem), and an important example in this class includes the sunflower spiral.Recent works by Akiyama, Marklof and Yudin provide a reasonable complete characterisation of planar spiral sets which are also Delone. A related problem that has emerged in several places in the literature over the past few years is to determine whether this theory can be extended to higher dimensions, and in particular to show the existence of spiral Delone sets in any dimension. This paper addresses this question by characterising the Delone property of a spiral set in terms of packing and covering conditions satisfied by the spherical sequence $(\mathbf{u}_k)_{k\ge 1}$. This allows for the construction of explicit examples of spiral Delone sets in $\mathbb{R}^n$ for all $n\ge 2$, which boils down to finding a sequence of points in $\mathbb{S}^{n-1}$ enjoying some optimal distribution properties.

The Schur covers of a dihedral group of $2$-power order are a dihedral group,a semi-dihedral group, and a generalized quaternion group.We prove that if one of the three groups is realizable as Galois group, thenother two groups are also realizable as Galois groups under a~condition on the existence of a certain quartic cyclic extension over the base field.

Let $f$ be a primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$. In this paper, we study the higher moments of general divisor problem related to the coefficients of Rankin-Selberg $L$-function $L(f\times f,s)$ associated with $f$ over sum of two squares.

We give a correspondence between automorphic pairs of distributions on $\mathbb{R}$ and Dirichlet series satisfying functional equations and some additional analytic conditions. Moreover, we show that the notion of automorphic pairs of distributions on $\mathbb{R}$ can be regarded as a generalization of automorphic distributions on smooth principal series representations of the universal covering group of $SL(2,\mathbb{R})$. As an application, we prove Weil type converse theorems for automorphic distributions and Maass forms of real weight.