Related papers: Drinfeld Modules with Complex Multiplication, Hass…
In this article we construct the arithmetic Satake compactification of the Drinfeld moduli schemes of arbitrary rank over the ring of integers of any global function field away from the level structure, and show that the universal family…
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…
Designing a deterministic polynomial time algorithm for factoring univariate polynomials over finite fields remains a notorious open problem. In this paper, we present an unconditional deterministic algorithm that takes as input an…
Polynomial factorization over $ZZ$ is of great historical and practical importance. Currently, the standard technique is to factor the polynomial over finite fields first and then to lift to integers. Factorization over finite fields can be…
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, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…
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…
Let $f(x)$ be a separable polynomial over a local field. Montes algorithm computes certain approximations to the different irreducible factors of $f(x)$, with strong arithmetic properties. In this paper we develop an algorithm to improve…
We set new speed records for multiplying long polynomials over finite fields of characteristic two. Our multiplication algorithm is based on an additive FFT (Fast Fourier Transform) by Lin, Chung, and Huang in 2014 comparing to previously…
Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. First, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope…
The factorization of the universal R-matrix corresponding to so called Drinfeld Hopf structure is described on the example of quantum affine algebra $U_q(\hat{sl}_2)$. As a result of factorization procedure we deduce certain differential…
We present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^{1+\epsilon(d)} \times (\log q)^{5+\epsilon(q)}$…
We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the…
We propose a polynomial time $f$-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$) for computing an isomorphism (if there is any) of a finite dimensional…
We study the family of irreducible modules for quantum affine $\lie{sl}_{n+1}$ whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the…
Rank-2 Drinfeld modules are a function-field analogue of elliptic curves, and the purpose of this paper is to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves in terms of supersingularity.…
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…
The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a…
We present a new algorithm for computing $m$-th roots over the finite field $\F_q$, where $q = p^n$, with $p$ a prime, and $m$ any positive integer. In the particular case $m=2$, the cost of the new algorithm is an expected $O(\M(n)\log (p)…