Title: Euler factors of equivariant $L$--functions of Drinfeld modules and beyond Show Chinese title

Title: Euler因子的等变$L$-函数和更广泛的德林舍尔模

Abstract: In \cite{FGHP}, the first author and his collaborators proved an equivariant Tamagawa number formula for the special value at $s=0$ of a Goss--type $L$--function, equivariant with respect to a Galois group $G$, and associated to a Drinfeld module defined on $\Bbb F_q[t]$ and over a finite, integral extension of $\Bbb F_q[t]$. The formula in question was proved provided that the values at $0$ of the Euler factors of the equivariant $L$--function in question satisfy certain identities involving Fitting ideals of certain $G$--cohomologically trivial, finite $\Bbb F_q[t][G]$--modules associated to the Drinfeld module. In \cite{FGHP}, we prove these identities in the particular case of the Carlitz module. In this paper, we develop general techniques and prove the identities in question for arbitrary Drinfeld modules. Further, we indicate how these techniques can be extended to the more general case of higher dimensional abelian $t$--modules, which is relevant in the context of the proof of the equivariant Tamagawa number formula for abelian $t$--modules given by N. Green and the first author in \cite{Green-Popescu}. This paper is based on a lecture given by the first author at ICMAT Madrid in May 2023 and builds upon results obtained by the second author in his PhD thesis \cite{Ramachandran-thesis}. Show Chinese abstract

Abstract: 在\cite{FGHP}中，第一作者和他的合作者证明了一个与 Goss 类型的$L$-函数在$s=0$处的特殊值相关的等变 Tamagawa 数公式，该公式相对于一个伽罗瓦群$G$是等变的，并且与定义在$\Bbb F_q[t]$上并在$\Bbb F_q[t]$的有限整扩张上的德林费尔德模相关。 所涉及的公式在满足所涉等变$L$-函数的Euler因子在$0$处的值满足某些涉及特定$G$-上同调平凡、有限$\Bbb F_q[t][G]$-模的Fitting理想的情况下被证明，这些模与Drinfeld模相关联。在\cite{FGHP}中，我们证明了Carlitz模的特殊情况下的这些恒等式。在本文中，我们发展了一般技术，并证明了任意Drinfeld模的相关恒等式。 此外，我们说明这些技术如何扩展到更高维阿贝尔$t$--模的一般情况，这在N. Green和第一作者在\cite{Green-Popescu}中给出的等变塔马格纳数公式的阿贝尔$t$--模背景下是相关的。 本文基于第一作者于2023年5月在马德里ICMAT所做的讲座，并建立在第二作者在其博士论文\cite{Ramachandran-thesis}中获得的结果之上。