AI 中文总结
该研究证明了特征p函数域上，Drinfeld模与Artin表示对应的扭曲特殊L值在所有非负整数处的超越性，并由此推导出另一类特殊L值的超越性。
AI 中文摘要
本文研究特征p函数域上取值的特定Goss型L级数在非负整数处特殊值的超越性。我们证明，对于K上的Drinfeld模φ及Artin表示ρ:G_K→GL_n(overline{F}_q)，对每个非负整数k，扭曲特殊值L(φ^∨,ρ,k)在K上是超越的。证明运用了Drinfeld模的Artin扭转理论、t-模的Taelman调节器，以及Gezmis和Namoijam关于对数可处理坐标的代数独立性定理。由此推论，对每个正整数k，特殊L值L(ρ,k)具有超越性。
英文摘要
In this article, we study the transcendence of special values of certain Goss type $L$-series at non-negative integers, which takes values in a function field of characteristic $p$. We show that, for a Drinfeld module $φ$ over $K$ and an Artin representation $ρ:G_K\to \operatorname{GL}_n(\overline{\mathbb{F}}_q)$, the twisted special value $L(φ^\vee,ρ,k)$ is transcendental over $K$ for every non-negative integer $k$. The proof uses the theory of Artin twists of Drinfeld modules, Taelman's regulators of $t$-modules, and the algebraic independence theorem of Gezmis and Namoijam for tractable coordinates of logarithms. As a consequence, we deduce the transcendence of the special $L$-value $L(ρ,k)$ for every positive integers $k$.
Comments31 pages
