Level-Structures of Drinfeld-Modules -- Closing a small gap

Equidistribution and integral points for Drinfeld modules

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 Siegel's integral points theorem over number fields and to an analog of a theorem of Schinzel's regarding the order of a point modulo certain primes.

Towards the full Mordell-Lang conjecture for Drinfeld modules

Let $ϕ$ be a Drinfeld module of generic characteristic, and let $X$ be a sufficiently generic affine subvariety of $\mathbb{G}_a^g$. We show that the intersection of $X$ with a finite rank $ϕ$-submodule of $\mathbb{G}_a^g$ is finite.

A Weil-Barsotti formula for Drinfeld modules

We study the group of extensions in the category of Drinfeld modules and Anderson's t-modules, and we show in certain cases that this group can itself be given the structure of a t-module. Our main result is a Drinfeld module analogue of the Weil-Barsotti formula for abelian varieties. Extensions of general t-modules are also considered, in particular extensions of tensor powers of the Carlitz module. We motivate these results from various directions and compare to the situation of elliptic curves.

用于Drinfeld模和Drinfeld模曲线的Pila-Zannier策略

The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves

研究将Pila-Zannier策略扩展到Drinfeld模，证明了相关定理类似物。在特征零中，用刚性解析Pila-Wilkie定理替代基于\(o\)-极小性的计数步骤，通过点数论证建立函数超越性输入，实现该策略在函数域的应用。

Differential Characters of Drinfeld Modules and de Rham Cohomology

We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical $F$-crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This $F$-…

Torsion of Drinfeld modules and equicharacteristic unimodular Galois covers

We prove that the groups PSL(r,q^d) can be realized F_q(T)-regularly as Galois groups over the purely transcendental field F_q(T)(t_1,...,t_{r-1}) if r is even and r/2 is coprime to q^d-1. The method is to use twisted moduli schemes of Drinfeld modules (a la Shih). If one has sufficient control over modular forms, it can be made into an algorithm to find explicit equations. We do this for PSL(2,q^2) using Drinfeld modular forms.

Invitation to higher local fields, Part II, section 4: Drinfeld modules and local fields of positive characteristic

This work is a survey of relations between Drinfeld modules and higher dimensional fields of positive characteristic. The main new result stated is the expression of vanishing orders of certain modular forms through partial zeta values.

Algebraic relations among periods and logarithms of rank 2 Drinfeld modules

For any rank 2 Drinfeld module rho defined over an algebraic function field, we consider its period matrix P, which is analogous to the period matrix of an elliptic curve defined over a number field. Suppose that the characteristic of F_q is odd and rho is without complex multiplication. We show that the transcendence degree of the field generated by the entries of P over F_q(theta) is 4. As a consequence, we show also the algebraic independence of Drinfeld logarithms of algebraic functions which are linearly independent over F_q(theta).

Galois表示对几乎所有Drinfeld模都是满射的

Galois representations are surjective for almost all Drinfeld modules

本文证明在函数域上，对于秩r≥2的Drinfeld模，T-adic Galois表示在密度为1的模集合上是满射的，利用Hilbert不可约性、Drinfeld均匀化理论和筛法。

Drinfeld modular curve and Weil pairing

In this paper we describe the compactification of the Drinfeld modular curve. This compactification is analogous to the compactification of the classical modular curve given by Katz and Mazur. We show how the Weil pairing on Drinfeld modules that we defined in earlier work gives rise to a map on the Drinfeld modular curve. We introduce the Tate-Drinfeld module and show how this describes the formal neighbourhood of the scheme of cusps of the Drinfeld modular curve.

Endomorphism Rings and Isogenies Classes for Drinfeld Modules of Rank 2 Over Finite Fields

Let $Φ$ be a Drinfeld $\mathbf{F}_{q}[T]$-module of rank 2, over a finite field $L$, a finite extension of $n$ degrees of a finite field with $q$ elements $\mathbf{F}_{q}$. Let $m$ be the extension degrees of $ L$ over the field $\mathbf{F}_{q}[T]/P$, $P$ is the $\mathbf{F}%_{q}[T]$-characteristic of $L$, and $d$ the degree of the polynomial $P$. We will discuss about a many analogies points with elliptic curves. We start by the endomorphism ring of a Drinfeld $\mathbf{F}_{q}[T]$-module of rank 2, End$_{L}Φ$, and we specify the maximality conditions and non maximality conditions as a $\mathbf{…

椭圆曲线与 Drinfeld 模的统一定义

Pour une définition commune des courbes elliptiques et modules de Drinfeld

本文为椭圆曲线与 Drinfeld 模提出统一代数定义，引入“模 élémentaires”并给出其分类，将两者类比提升为统一理论。

计数满足指定局部条件的Drinfeld模

Counting Drinfeld modules with prescribed local conditions

该研究针对$\mathbb{F}_q(T)$上给定高度且满足指定局部条件的Drinfeld $\mathbb{F}_q[T]$-模数量给出渐近估计，通过关联加权射影空间点计数问题实现，其关于全局函数域上加权射影空间有界高度点计数的结果具独立意义。

Anneaux d'Endomorphismes de modules de Drinfeld de rang 2

We discuss many analogy points with the elliptic curves. more precisely, we study the characteristic polynomial of a Drinfeld module of rank 2 and use it to calculate the number of isogeny classes for such modules.

Can a Drinfeld module be modular?

Let $k$ be a global function field with field of constants $\Fr$ and let $\infty$ be a fixed place of $k$. In his habilitation thesis \cite{boc2}, Gebhard Böckle attaches abelian Galois representations to characteristic $p$ valued cusp eigenforms and double cusp eigenforms \cite{go1} such that Hecke eigenvalues correspond to the image of Frobenius elements. In the case where $k=\Fr(T)$ and $\infty$ corresponds to the pole of $T$, it then becomes reasonable to ask whether rank 1 Drinfeld modules over $k$ are themselves ``modular'' in that their Galois representations arise from a cusp or double…

Monodromy Groups associated to Non-Isotrivial Drinfeld Modules in Generic Characteristic

Let $ϕ$ be a non-isotrivial family of Drinfeld A-modules of rank r in generic characteristic with a suitable level structure over a connected smooth algebraic variety X. Suppose that the endomorphism ring of $ϕ$ is equal to A. Then we show that the closure of the analytic fundamental group of X in $SL_r(\mathbb{A}_F^f)$ is open, where $\mathbb{A}_F^f$ denotes the ring of finite adeles of the quotient field F of A. From this we deduce two further results: (1) If X is defined over a finitely generated field extension of F, the image of the arithmetic étale fundamental group of X on the adelic Ta…

The A-module Structure Induced by a Drinfeld A-module over a Finite Field

Let $Φ$ 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 $P_Φ(X)=$ $X^{2}-cX+μP^{m}$ ($c$ an element of $% \mathbf{F}_{q}[T]$ and $μ$ a no null element of $\mathbf{F}_{q}$, $m$ the degree of the extension $L$ over the field $\mathbf{F}_{q}[T]/P$, $P$ is a $\mathbf{F}%_{q}[T]$-characteristic of $L$ and $d$ the degree of the polynomial $P$) the characteristic polynomial, of the Frobenius $F$ of $L$. We will interested to the structure of finite $\mathbf{F}_{q}[T]$-module $L^…

椭圆曲线上的秩二Drinfeld模

Rank-Two Drinfeld Module over Elliptic Curves

本文显式构造了椭圆曲线坐标环上的秩二符号正规化Drinfeld模，给出了其Anderson motive的结构和二次τ-关系，并确定了模空间的显式方程。

Statistique sur la cyclicité de A-module de Drinfeld de rang 2

Let $Φ$ be a Drinfeld $\mathbf{F}\_{q}[T]$-module of rank 2, over a finite field $L=\mathbf{F}\_{q^{n}}$. We will study the cyclic property of the structure $L^Φ.$ We will prove that the latter is cyclic only for trivial extensions of $\mathbf{F}\_{q}$.

Drinfeld 模的模空间的环形紧化

Toroidal compactifications of the moduli spaces of Drinfeld modules

本文通过吹胀 Pink 的 Satake 紧化并结合形式模理论与迭代 Tate 一致化，构造了秩 d、带级 N 结构的 Drinfeld 模空间的对数正则环形紧化，并证明其中存在正则紧化。

Beilinson-Drinfeld Schubert varieties and global Demazure modules

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson- Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra.

Algebraic equations on the adelic closure of a Drinfeld module

Let $k$ be a field of positive characteristic and $K = k(V)$ a function field of a variety $V$ over $k$ and let ${\mathbf A}_K$ be a ring of adéles of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $Φ:{\mathbb F}[t] \to \operatorname{End}_K({\mathbb G}_a)$ over $K$ and a positive integer $g$ we regard both $K^g$ and ${\mathbf A}_K^g$ as $Φ({\mathbb F}_p[t])$-modules under the diagonal action induced by $Φ$. For $Γ\subseteq K^g$ a finitely generated $Φ(\F_p[t])$-submodule and an affine subvariety $X \subseteq \bG_a^g$ define…

Sur la Structure de A-module de Drinfeld de rang 2

$Φ$ be a Drinfeld $\mathbf{F}\_{q}[T]$-module of rank 2, over a finite field $L$. Let $P\_Φ(X)=$ $X^{2}-cX+μP^{m}$ ($c$ an element of $\mathbf{F}\_{q}[T],$ $μ$ be a non-vanishing element of $% \mathbf{F}\_{q}$, $m$ the degree of the extension $L$ over the field $% \mathbf{F}\_{q}[T]/P,$ and $P$ the $\mathbf{F}\_{q}[T]$-characteristic of $% L $ and $d $ the degree of the polynomial $P$) the characteristic polynomial of the Frobenius $F$ of $L$. We will be interested in the structure of finite $\mathbf{F}\_{q}[T]$-module $L^Φ$ induced by $Φ$ over $L$. Our main result is analogue to that of Deuri…

Beilinson-Drinfeld Grassmannian上的Langlands对偶

Langlands duality on the Beilinson-Drinfeld Grassmannian

本研究用Langlands对偶术语计算Beilinson-Drinfeld Grassmannian上的等变层范畴，得到可因子化导出几何Satake定理，还算出对应无分歧向量的范畴类比，其谱范畴涉及Langlands对偶群相关因子化代数的因子化模。