Related papers: Anneaux d'Endomorphismes de modules de Drinfeld de…

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…

We study tensor powers of rank 1 sign-normalized Drinfeld A-modules, where A is the coordinate ring of an elliptic curve over a finite field. Using the theory of A-motives, we find explicit formulas for the A-action of these modules. Then,…

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 describe moduli spaces of logarithmic rank $2$ connections on elliptic curves with $n \geq 1$ poles and generic residues. In particular, we generalize a previous work by the first and second named authors. Our main approach is to analyze…

In this paper, we compute the natural density of rank-$1$ Drinfeld module over $\mathbb{F}_q[T]$ with surjective adelic Galois representation; and the natural density of rank-$2$ Drinfeld modules over $\mathbb{F}_q[T]$ whose…

We study rational points and torsion points on Drinfeld modular curves defined over rational function fields. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two over $\Bbb…

Let $\mathbb{F}_{q}$ be a finite field, and $A:=\mathbb{F}_{q}[T]$. In this article, we give explicit criteria, involving concrete valuations, on the coefficients of the Drinfeld $A$-modules of rank $r$ for $r=2,3$, which ensure the…

We study the modular curves defined by Weber functions, and associated modular polynomials, action of $\mathrm{SL}_2(\mathbb{Z})$, and parametrizations of elliptic curves with a view to the study of the isogeny graphs that they determine,…

In analogy with the periods of abelian integrals of differentials of third kind for an elliptic curve defined over a number field, we introduce a notion of periods of third kind for a rank 2 Drinfeld Fq[t]-module rho defined over an…

In this paper, we study the surjectivity of adelic Galois representation associated to Drinfeld $\mathbb{F}_q[T]$-modules over $\mathbb{F}_q(T)$ of rank $2$ in the cases when $q$ is even or $q=3$.

We study genus 2 function fields with elliptic subfields of degree 2. The locus $\L_2$ of these fields is a 2-dimensional subvariety of the moduli space $\mathcal M_2$ of genus 2 fields. An equation for $\L_2$ is already in the work of…

We present an algorithm solving the following problem: given two genus 2 curves over a field k with isogenous Jacobians, compute such an isogeny explicitly. This isogeny can be either an l-isogeny or, in the real multiplication case, an…

We describe the moduli space of logarithmic rank 2 connections on elliptic curves with 2 poles.

In this paper, we develop a view of self-isogenous modular polynomials and the $\mathfrak{l}$-cyclic isogeny graph for CM Drinfeld modules of arbitrary rank $r$. On the computational side, we give an explicit procedure to construct the…

This is the second of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present part, we compare the analytic theory with the algebraic one that was begun in a paper of the third…

We compute the first and second moments of the divisor-counting function for the Euler-Poincar\'{e} characteristic and the trace of Frobenius for the reductions modulo $p$ of a rank 2 Drinfeld module with nontrivial endomorphism ring, as…

We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for $\mathrm{GL}_2(F)$, for its subgroups of matrices whose determinant has even (respectively trivial)…

We consider the Siegel modular variety of genus 2 and a p-integral model of it for a good prime p>2, which parametrizes principally polarized abelian varieties of dimension two with a level structure. We consider cycles on this model which…

Let q be a prime power and E a non-isotrivial elliptic curve over Fq(T) given by a Weierstrass model. We survey the construction, with an explicit point of view, of the modular parametrization of E by the associated Drinfeld modular curve.…

Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in $\mathbf{F}_q[t]$. In this paper, we define an analogue of the Hodge-Tate map which is suitable for the study of Drinfeld…