论文标题
Drinfeld模块和应用
An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications
论文作者
论文摘要
我们修复了数据$（k/f，e）$，由galois扩展名$ k/f $ p $ p $ p $ p $全球字段，带有任意的阿贝利安galois $ g $和一个德林菲尔德模块$ e $ $ e $，该$ e $ $ e $定义了$ f $的某些dedekind子。对于此数据，我们定义A $ G $ -Equivariant $ l $ -function $θ_{k/f}^e $ $，并证明了其特殊值$θ_{k/f}^e（0）$的某些Euler填写版本的equivariant tamagawa编号公式。这将taelman的班级编号公式用于goss zeta函数的$ζ_F^e（0）$ $ζ_F^e $与Pair $（F，e）$相关的$ζ_F^e $。 Taelman的结果是通过设置$ k = f $从我们的结果中获得的。结果，我们证明了经典（数字）精制brumer的完美德林菲尔德模块类似物 - stark猜想，将taelman的类别组$ h（e/k）$与特殊值$θ_{k/f}^e（0）$相关联。
We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a $G$-equivariant $L$-function $Θ_{K/F}^E$ and prove an equivariant Tamagawa number formula for certain Euler-completed versions of its special value $Θ_{K/F}^E(0)$. This generalizes Taelman's class number formula for the value $ζ_F^E(0)$ of the Goss zeta function $ζ_F^E$ associated to the pair $(F, E)$. Taelman's result is obtained from our result by setting $K=F$. As a consequence, we prove a perfect Drinfeld module analogue of the classical (number field) refined Brumer--Stark conjecture, relating a certain $G$-Fitting ideal of Taelman's class group $H(E/K)$ to the special value $Θ_{K/F}^E(0)$ in question.
