关于秩为2的Drinfeld A-模所附带的Galois表示的满射性
On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$
AI总结：
本文研究了秩为2的Drinfeld A-模所附带的Galois表示的满射性，通过水平和垂直方向的方法给出条件并获得新的例子，证明了在特定条件下表示的满射性。
摘要：
令F_q为具有q个元素的有限域，其中q是素数幂，令A:=(F_q)[T]。根据PR09，秩为2的Drinfeld A-模φ所附带的Galois表示的adelic图像为开集，确定其是否满射仍是一个微妙的问题。为解决此问题，本文研究了φ所附带的Galois表示的p-adic满射性，其中p∈Ω_A:Spec(A)\{(0)\}。研究此问题有两个方向：一个固定素数p，另一个固定φ。在水平方向上，对于固定的素数p∈Ω_A，我们给出了Drinfeld A-模φ的秩为2的明确且易于验证的条件，以确保p-adic Galois表示ρ_{φ,p}的满射性。这项工作不仅扩展了Ray24对于p=(T)的工作，还获得了在相对更简单的条件下的一种变体。在垂直方向上，我们证明对于固定秩为2的Drinfeld A-模φ，其系数满足特定的同余和赋值条件时，对于所有p∈Ω_A，Galois表示ρ_{φ,p}是满射的。这恢复了Zyw11的例子，并获得了超出Zyw25考虑范围的新例子。作为结果，我们获得了所附带的adelic Galois表示的满射性。
英文摘要：
Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is a prime power and let $A:= \mathbb{F}_{q}[T]$. By~\cite{PR09}, the adelic image of the Galois representation attached to a rank $2$ Drinfeld $A$-module $φ$ is open, and determining when it is surjective remains a subtle problem. To resolve this question, in this article, we study the $\mathfrak{p}$-adic surjectivity of the Galois representations attached to $φ$, where $\mathfrak{p} \in Ω_A:= \mathrm{Spec}(A) \setminus \{ (0) \}$. There are two directions to investigate this problem: one by fixing the prime $\mathfrak{p}$, and the other by fixing $φ$. In the horizontal direction, for a fixed prime $\mathfrak{p} \in Ω_A$, we give explicit and easily verifiable conditions on Drinfeld $ A$-modules $φ$ of rank $2$ which ensure the surjectivity of the $\mathfrak{p}$-adic Galois representation $ρ_{φ,\mathfrak{p}}$. This work not only extends the work of~\cite{Ray24} for $\mathfrak{p}=(T)$, but also obtains a variant of~\cite{Ray24} under comparatively simpler conditions in the case $\mathfrak{p}=(T)$. In the vertical direction, we show that for a fixed rank $2$ Drinfeld $A$-module $φ$, whose coefficients satisfy certain congruence and valuation conditions, the $\mathfrak{p}$-adic Galois representation $ρ_{φ,\mathfrak{p}}$ is surjective for all primes $\mathfrak{p} \in Ω_A$. This recovers the example of \cite{Zyw11} and yields new examples beyond those considered in \cite{Zyw25}. As a consequence, we obtain the surjectivity of the associated adelic Galois representation.
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