Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 1980s where they were at the heart of comparison isomorphisms, further important applications, e.g., to transcendence theory have only been discovered recently. The Anderson–Thakur special function interpolates L -values via Pellarin-type identities, and its values at algebraic elements recover Gauss–Thakur sums, as shown by Anglès and Pellarin. For Drinfeld–Hayes modules, generalizations of Anderson generating functions have been introduced by Green–Papanikolas and – under the name of “special functions” – by Anglès, Ngo Dac and Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A -module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A -motive. We also undertake the study of Gauss–Thakur sums for Anderson A -modules, and show that the result of Anglès–Pellarin relating values of the special functions to Gauss–Thakur sums holds in this generality.

中文翻译：

---

特殊函数和高斯-塔库尔求和在更高的等级和维度上

近年来，安德森生成函数在函数域算术中受到越来越多的关注。尽管它们是比较同构的核心，但安德森（Anderson）在1980年代引入了它们，但更重要的应用（例如，超越理论）直到最近才被发现。Anderson–Thakur特殊函数通过Pellarin类型恒等式对L值进行插值，并且其在代数元素上的值恢复了Gauss–Thakur和，如Anglès和Pellarin所示。对于Drinfeld-Hayes模块，Green-Papanikolas引入了安德森生成函数的泛化，Anglès，Ngo Dac和Tavares Ribeiro则以“特殊函数”的名义引入了泛化。在本文中，我们提供了附加到任何Anderson A模块的特殊功能的一般构造。我们展示了特殊功能空间与周期晶格以及与A动机的贝蒂同调的直接联系。我们还对安德森A-模块的高斯-塔库尔和进行了研究，并证明了将特殊函数的值与高斯-塔库尔和相关的Anglès-Pellarin的结果在这种概论中成立。