Related papers: On field extensions given by periods of Drinfeld m…

This is the first of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank r. In the present part, we develop the analytic theory. Most of the work goes into defining and studying the…

In this paper, we study the ramification of extensions of a function field generated by division points of rank 2 Drinfeld modules. Also conductors of certain rank 2 Drinfeld modules are defined as analogues of those for elliptic curves. A…

We present a new notion of distribution and derived distribution of rank $r \in \mathbb{N}$ for a global function field $K$ with a distinguished place $\infty$. It allows to describe the relations between division points, isogenies, and…

Let $K$ be a totally real number field, $d$ a positive integer, and $Q$ a higher degree form over $K$. We prove that there are at most finitely many totally real extensions $L/K$ of degree $d$ such that $Q$ over $L$ is universal. Further,…

The modular invariant of rank 1 Drinfeld modules is introduced and used to formulate and prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is a version of Shimura's Main Theorem…

We give explicit bounds for Zsigmondy sets of certain families of Drinfeld modules of rank 2. The primary strategy is to bound the local heights associated to Drinfeld modules and then relate canonical to classical heights.

Let $G$ be a finite group and $K$ a number field. We construct a $G$-extension $E/F$, with $F$ of transcendence degree $2$ over $K$, that specializes to all $G$-extensions of $K_\mathfrak{p}$, where $\mathfrak{p}$ runs over all but finitely…

We give a lower bound for the local height of a non-torsion element of a Drinfeld module.

We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant less than X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the…

The primary objective of this paper is to derive explicit formulas for rank one and rank two Drinfeld modules over a specific domain denoted by A. This domain corresponds to the projective line associated with an infinite place of degree…

Let $\Phi $ be a Drinfeld $\mathbf{F}_{q}[T]$-module of rank 2, over a finite field $L$, a finite extension of $n$ degrees for a finite field with $q$ elements $% \mathbf{F}_{q}$. Let $P_{\Phi}(X)=$ $X^{2}-cX+\mu P^{m}$ ($c$ an element of…

In this paper we provide a short proof of the Riemann Hypothesis for Drinfeld modules which uses only basic notions from the theory of global function fields and of Drinfeld modules.

A new proof of an old theorem of Drinfeld concerning the representability of the moduli problem of special formal $\mathcal{O}_{D}$-modules by Deligne's $p$-adic formal model of Drinfeld's upper half-plane is given for $d=2.$ The display…

Let $q\geq2$ be a prime power and consider Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$. We prove that there are no points with coordinates being Drinfeld singular moduli, on a family of hyperbolas $XY=\gamma$, where $\gamma$ is a…

In this note, we continue to be interested in the relationship that connects the restricted distribution of finitude at the local level of intermediate fields of a purely inseparable extension $K/k$ to the absolute or global finitude of…

It is proved that Ulrich modules exist for a large class of local rings of dimension two. This complements earlier work of the authors and Ziquan Zhuang that described complete intersection domains of dimension two that admit no Ulrich…

We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic…

In this paper, we prove a generalization of Green's Hyperplane Restriction Theorem to the case of modules over the polynomial ring, providing in particular an upper bound for the Hilbert function of the general linear restriction of a…

We study the universality of forms of degrees greater than 2 over rings of integers of totally real number fields. We show that such universal forms always exist, but cannot be characterized by any variant of the 290-Theorem of…

We prove an analog of Siegel's theorem for integral points in the context of Drinfeld modules. The result holds for finitely generated submodules of the additive group over a function field of transcendence dimension 1.