论文标题
T木川的t木川编号公式用于T模型和应用
An Equivariant Tamagawa Number Formula for t-Modules and Applications
论文作者
论文摘要
我们修复了动机数据$（k/f，e）$，由galois扩展名$ k/f $ p $ p $ p $ p $ p $ p $ p $ p $全球田地，带有任意的abelian galois $ g $和ableian $ t $ t $ -module $ e $，定义在某些dedekind子$ f $的$ f $上。对于此数据，可以定义A $ G $ -Equivariant Motivic $ L $ -Function $θ_{K/F}^E $。我们完善了作者先前作品中开发的技术，并证明了tamagawa数字公式，用于适当的Euler产品的特殊值$θ_{k/f}^e（0）$（0）$ l $ l $ unction。这将作者的先前结果从德林菲尔德模块设置扩展到$ t $ - 模块设置。首先值得注意的是，我们证明了古典（数字字段）完善的Brumer-Stark猜想的$ T $模块类似物，将Taelman类模型的$ T $类似物$ h（e/k）的特定$ g $适合的理想与特殊价值$θ_{k/f}^e（0）$（0）$ qood。第二个结果，我们证明了值$θ_{k/f}^e（m）$的公式，当$ e $是drinfeld模块时，所有正整数$ m \ in \ bbb z _ {\ geq 0} $。 This, in turn, implies a Drinfeld module analogue of the classical Refined Coates-Sinnott Conjecture relating $Θ_{K/F}^E(m)$ to the Fitting ideals of certain Carlitz twists $H(E(m)/\mathcal O_K)$ of Taelman's class modules, suggesting a strong analogy between these twists and the even Quillen $K$-groups of a number 场地。在即将发表的论文中，这些后果将用于开发泰勒曼类模块的$ t $模块类似物的硫磺理论。
We fix motivic data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and an ableian $t$-module $E$, defined over a certain Dedekind subring of $F$. For this data, one can define a $G$-equivariant motivic $L$-function $Θ_{K/F}^E$. We refine the techniques developed in previous work of the authors and prove an equivariant Tamagawa number formula for appropriate Euler product completions of the special value $Θ_{K/F}^E(0)$ of this equivariant $L$-function. This extends previous results of the authors from the Drinfeld module setting to the $t$--module setting. As a first notable consequence, we prove a $t$-module analogue of the classical (number field) Refined Brumer-Stark Conjecture, relating a certain $G$-Fitting ideal of the $t$-motive analogue $H(E/K)$ of Taelman's class modules to the special value $Θ_{K/F}^E(0)$ in question. As a second consequence, we prove formulas for the values $Θ_{K/F}^E(m)$, at all positive integers $m\in\Bbb Z_{\geq 0}$, when $E$ is a Drinfeld module. This, in turn, implies a Drinfeld module analogue of the classical Refined Coates-Sinnott Conjecture relating $Θ_{K/F}^E(m)$ to the Fitting ideals of certain Carlitz twists $H(E(m)/\mathcal O_K)$ of Taelman's class modules, suggesting a strong analogy between these twists and the even Quillen $K$-groups of a number field. In an upcoming paper, these consequences will be used to develop an Iwasawa theory for the $t$-module analogues of Taelman's class modules.
