Exceptional cases of adelic surjectivity for Drinfeld modules of rank 2

Finding Endomorphisms of Drinfeld modules

We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof.

椭圆曲线和Drinfeld模的挠界

Torsion bounds for elliptic curves and Drinfeld modules

我们得到了定义在一个有限生成域K上的给定椭圆曲线或Drinfeld模上的L-有理扭点个数的渐近最优上界，它是次数[L：K]的函数.我们的主要工具是adelic开放的形象伽罗瓦表示重视椭圆曲线和德林费尔德模块，由于塞尔和粉红Ruetsche，分别。我们的方法是证明某些伽罗瓦模的一般结果，它同时适用于椭圆曲线和Drinfeld模。

Surjectivity of Galois Representations in Rational Families of Abelian Varieties

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g \geq 3$, there are infinitely many abelian varieties over $\mathbb Q$ with adelic Galois representation havi…

具有大Galois像的秩-2 Drinfeld模的自然密度

Natural density of rank-$2$ Drinfeld modules with big Galois image

本文计算了$\mathbb{F}_q[T]$上具有满射adelic Galois表示的秩-1 Drinfeld模，以及对于有限多个素理想$\mathfrak{l}$其Galois像包含特殊线性子群的秩-2 Drinfeld模的自然密度。

Drinfeld模的局部Kummer理论

Local Kummer theory for Drinfeld modules

本文研究了局部域上Drinfeld模的惯性群在修改后的adelic Tate模上的作用，通过典型局部Kummer配对描述惯性群作用像，给出了其有效公式及为开集的充要条件。

一类具有满射伽罗瓦表示的秩3 Drinfeld $A$-模

A class of Drinfeld $A$-modules of rank $3$ with surjective Galois representations

本文针对奇素数幂$q=p^e\geq7$的情形，构造了含双参数的无穷多秩3 Drinfeld $A$-模，证明其对任意非零素理想的各类伽罗瓦表示均为满射，推广了此前仅适用于$p\equiv1\pmod{3}$的具体例子。

类属特征的非等平凡Drinfeld模的单值群

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

设$ϕ$是连通光滑代数簇X上具有适当水平结构的类属特征秩为r的非等平凡Drinfeld A-模族.设$ϕ$的自同态环等于A.然后证明了X的解析基本群在$SL_r(\mathbb{A}_F^f)$中的闭包是开的，这里$\mathbb{A}_F^f$表示A的商域F的有限三角环.（1）若X定义在F的一个生成域扩张上，则X的算术代数基本群在$ϕ$的Adelic Tate模上的象在$GL_r(\mathbb{A}_F^f)$中是开的. (2)设$ψ$是定义在F的一个有限生成域扩张上的秩为r的Drinfeld A-模，并假设$ψ$不能定义在F的一个有限扩张上.再次假设$ψ$的自同态环是A。然后，在$GL_r(\mathbb{A}_F^f)$中开了$ψ$的Adelic Tate模上的伽罗瓦表示的象.最后，我们将上述结果推广到任意自同态环的情形。

A Hecke-equivariant decomposition of spaces of Drinfeld cusp forms via representation theory, and an investigation of its subfactors

There are various reasons why a naive analog of the Maeda conjecture has to fail for Drinfeld cusp forms. Focussing on double cusp forms and using the link found by Teitelbaum between Drinfeld cusp forms and certain harmonic cochains, we observed a while ago that all obvious counterexamples disappear for certain Hecke-invariant subquotients of spaces of Drinfeld cusp forms of fixed weight, which can be defined naturally via representation theory. The present work extends Teitelbaum's isomorphism to an adelic setting and to arbitrary levels, it makes precise the impact of representation theory,…

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。