Related papers: Lehmer problem and Drinfeld modules
Lower Bound for the Canonical Height for Drinfeld Modules with Complex Multiplication. Let K be a fi nite extension of Fq(T), let L=K be a Galois extension with Galois group G and let E be the sub eld of L fixed by the center of G. Assume…
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.
We provide explicit bounds on the difference of heights of isogenous Drinfeld modules. We derive a finiteness result in isogeny classes. In the rank 2 case, we also obtain an explicit upper bound on the size of the coefficients of modular…
We give a lower bound for the local height of a non-torsion element of a Drinfeld module.
We prove that the local height of a point on a Drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module. This gives rise to a Drinfeld module analog of a weak version of…
In this paper, we study the Kummer pairing associated with formal Drinfeld modules having stable reduction of height one. We give an explicit description of the pairing \`a la Kolyvagin, in terms of the logarithm of the formal Drinfeld…
In this paper, we generalize Dorman's work to estimate singular moduli for higher rank Drinfeld modules. In particular, we give a lower bound on the valuation of singular moduli for Drinfeld modules with complex multiplication by an…
In this paper, we prove explicit reciprocity laws for a class of formal Drinfeld modules having stable reduction of height one, in the spirit of those existing in characteristic zero (cf. the work of Wiles). We begin by defining the Kummer…
This survey provides a practical and algorithmic perspective on Drinfeld modules over $\mathbb F_q[T]$. Starting with the construction of the Carlitz module, we present Drinfeld modules in any rank and some of their arithmetic properties.…
We obtain explicit upper and lower bounds on the size of the coefficients of the Drinfeld modular polynomials $\Phi_N$ for any monic $N\in\mathbb{F}_q[t]$. These polynomials vanish at pairs of $j$-invariants of Drinfeld…
In this paper we prove a special case of the Lehmer inequality for Drinfeld modules. Also, based on this inequality, we prove certain Mordell-Weil type of theorems for certain infinitely generated fields.
Let $q$ be an odd number and $q>5$, and $\mathbb{F}_q$ be a finite field of $q$ elements. We prove that at most finitely many singular moduli of rank 2 $\mathbb{F}_q[t]$-Drinfeld modules are algebraic units. In particular, we develop some…
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…
We provide two families of algorithms to compute characteristic polynomials of endomorphisms and norms of isogenies of Drinfeld modules. Our algorithms work for Drinfeld modules of any rank, defined over any base curve. When the base curve…
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…
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…
We present novel algorithms to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial…
In the present paper, we introduce meromorphic Drinfeld modular forms of arbitrary rank equipped with a particular arithmeticity property. We also study their special values at CM points and show the algebraic independence of these values…
We present a novel randomized algorithm to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo…
We give asymptotics for the number of Drinfeld $\mathbb{F}_q[T]$-modules over $\mathbb{F}_q(T)$ of a given height, which satisfy prescribed sets of local conditions. This is done by relating our problem to a problem about counting points on…