L函数的可分性、多值算子和零点
Separability, multi-valued operators, and zeroes of L-functions
摘要：
设$\k$是$\Fp$的有限扩张上的一个单变量全局函数域，$p$素数，$\infty$是$\k$的一个固定位置，$\A$是$\k$正则于$\infty$之外的函数环.设$E$是一个Drinfeld模或$T$ -模。然后，如在\cite{go1}中那样，可以基于阿贝尔簇的经典模型构造关联的特征$p$ $L$ -函数{ \it once}，进行某些辅助选择。我们在本文中的目的是要显示如何著名的概念“最大可分（在完成$\k_\infty$）子域”允许一个建设从这样的$L$ -功能某些可分的扩展是独立于这些选择。然后，这些字段将仅依赖于原始$T$模块或Drinfeld模块和$y\in \Zp$的islogic类，并且应该可以用这些术语进行描述。此外，它们给出了一个非常有用的框架，在其中查看\cite{w1}、\cite{dv1}、\cite{sh1}的“黎曼假设"证据。我们还建立了一个元素，这是{ \it可分离}代数上$\k_\infty$可以实现为一个“多值算子”的一般$T$ -模。这非常类似于在$\C^\ast$上实现1/2作为多值运算符$x\mapsto \sqrt{x}$。简单的例子表明，对于不可分元素，这个结果是错误的.这一结果可能最终允许一个“两个$T$”的解释，上述扩展在$E$上的多值算子和某些张量扭曲。
英文摘要：
Let $\k$ be a global function field in 1-variable over a finite extension of $\Fp$, $p$ prime, $\infty$ a fixed place of $\k$, and $\A$ the ring of functions of $\k$ regular outside of $\infty$. Let $E$ be a Drinfeld module or $T$-module. Then, as in \cite{go1}, one can construct associated characteristic $p$ $L$-functions based on the classical model of abelian varieties {\it once} certain auxiliary choices are made. Our purpose in this paper is to show how the well-known concept of ``maximal separable (over the completion $\k_\infty$) subfield'' allows one to construct from such $L$-functions certain separable extensions which are independent of these choices. These fields will then depend only on the isogeny class of the original $T$-module or Drinfeld module and $y\in \Zp$, and should presumably be describable in these terms. Moreover, they give a very useful framework in which to view the ``Riemann hypothesis'' evidence of \cite{w1}, \cite{dv1}, \cite{sh1}. We also establish that an element which is {\it separably} algebraic over $\k_\infty$ can be realized as a ``multi-valued operator'' on general $T$-modules. This is very similar to realizing 1/2 as the multi-valued operator $x\mapsto \sqrt{x}$ on $\C^\ast$. Simple examples show that this result is false for non-separable elements. This result may eventually allow a ``two $T$'s'' interpretation of the above extensions in terms of multi-valued operators on $E$ and certain tensor twists.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
