论文标题
Stickelberger元素的完整性和自然GALOIS模块的歼灭
Integrality of Stickelberger elements and annihilation of natural Galois modules
论文作者
论文摘要
每个Galois扩展名$ l/k $的数字字段和Galois Group $ g $和每个整数$ r \ r \ leq 0 $一个人可以将stickelberger元素关联到理性组环$ \ mathbb {q} [q} [q} [q} [g] $中的artin $ l $ l $ - $ r $ $ r $的价值方面。我们表明，每当$ g $ nilpotent时，其系数的分母都受到$ g $的换向器子组$ g'$ g'$ g'$的基数。此外，我们表明，在乘以$ | g'| $并远离$ 2 $ - 主要零件之后，如果$ r = 0 $ = 0 $和更高的Quillen $ k $ - $ l $ l $ l $ l $ r $ r <0 $，则将$ l $ $ l $的班级歼灭。这概括了Brumer和Coates和Sinnott从Abelian到Nilpotent扩展的猜想的最新进展。 对于任意$ g $，我们表明分母仍沿循环元素$ \ mathbb {z} _p $ - $ l $的$ l $的限制。这使我们能够对格林伯格的问题和关于$ p $ adiC artin $ l $ l $ series的行为的肯定答案。
To each Galois extension $L/K$ of number fields with Galois group $G$ and each integer $r \leq 0$ one can associate Stickelberger elements in the centre of the rational group ring $\mathbb{Q}[G]$ in terms of values of Artin $L$-series at $r$. We show that the denominators of their coefficients are bounded by the cardinality of the commutator subgroup $G'$ of $G$ whenever $G$ is nilpotent. Moreover, we show that, after multiplication by $|G'|$ and away from $2$-primary parts, they annihilate the class group of $L$ if $r=0$ and higher Quillen $K$-groups of the ring of integers in $L$ if $r<0$. This generalizes recent progress on conjectures of Brumer and of Coates and Sinnott from abelian to nilpotent extensions. For arbitrary $G$ we show that the denominators remain bounded along the cyclotomic $\mathbb{Z}_p$-tower of $L$ for every odd prime $p$. This allows us to give an affirmative answer to a question of Greenberg and of Gross on the behaviour of $p$-adic Artin $L$-series at zero.
