Related papers: On singular moduli for higher rank Drinfeld module…

This survey discusses hyperbolicity properties of moduli stacks and generalisations of the Shafarevich Hyperbolicity Conjecture to higher dimensions. It concentrates on methods and results that relate moduli theory with recent progress in…

For an extension $K/\mathbb{F}_q(T)$ of the rational function field over a finite field, we introduce the notion of virtually $K$-rational Drinfeld modules as a function field analogue of $\mathbb{Q}$-curves. Our goal in this article is to…

Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum…

Inspired by the relations between periods of elliptic integrals of the third kind and the periods of the extensions of the corresponding elliptic curves by the multiplicative group, we introduce the notion of the third kind periods for…

Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms…

We show that the module of integral points on a Drinfeld module satisfies a an analogue of Dirichlet's unit theorem, despite its failure to be finitely generated. As a consequence, we obtain a construction of a canonical finitely generated…

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms…

It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating…

We give a global description of the Frobenius elements in the division fields of Drinfeld modules of rank $2$. We apply this description to derive a criterion for the splitting modulo primes of a class of non-solvable polynomials, and to…

We present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class…

We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank $2$ Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from…

Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, presented as a path algebra modulo relations; further, assume that $\Lambda$ is graded by lengths of paths. The paper addresses the classifiability, via…

We study the moduli space of solutions to the Seiberg-Witten equations with $N$ spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent…

We determine the combinatorics of transitive module categories over the monoidal category of finite dimensional $\mathfrak{sl}_3$-modules which arise when acting by the latter monoidal category on arbitrary simple $\mathfrak{sl}_3$-modules.…

We investigate Drinfeld modular polynomials parametrizing $T$-isogenies between Drinfeld $\mathbb{F}_q[T]$-modules of rank $r\geq 2$. By providing an explicit classification of such isogenies, we derive explicit bounds on the $T$-degrees of…

This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of…

It is often stated that the Carlitz module is to the ring of univariate polynomials over a finite field what the multiplicative group is to the ring of integers. This analogy extends to the "rank 2" case, where Drinfeld modules play a role…

The goal of this article is to define an analogue of the Weil-pairing for Drinfeld modules using explicit formulas and to deduce its main properties from these formulas. Our result generalizes the formula currently known for rank 2 Drinfeld…

We define the canonical submodule of a Drinfeld module of rank greater than one over the affine line over a finite field. (This extends the definition of the level 1 canonical subgroup of Hattori for rank 2 with ordinary reduction.) We give…

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the question…