arXiv 1402.54512014-02-21math.NTmath.KT

Hecke特征标与全实数域和CM数域的$K$ -理论

Hecke characters and the $K$-theory of totally real and CM number fields

Grzegorz Banaszak, Cristian D. Popescu

备注：: 38 pages

摘要：

设$F/K$是数域的阿贝尔扩张，$F$是CM或全实的，$K$是全实的。若$F$是CM，且Brumer-Stark猜想对$F/K$成立，我们构造了一类无穷型$F$的$G(F/K)$ --等变Hecke特征标，该特征标等于某些$G(F/K)$ --等变$L$ -函数的特殊值.利用Greither-Popescu关于Brumer-Stark猜想的结果，我们构造了这些特征的$l$ -adic非本原形式，对于素数$l> 2$。进一步，利用这些$l$ -adic Hecke特征标的特殊值，构造了$F$的$l$ -准Quillen局部化序列中的$G(F/K)$ -等变Stickelberg分裂映射，推广了Banaszak在1990年对$K = \Bbb Q$的结果.我们还应用Stickelberg分裂映射来构造$l$中的特殊元素-$K_{2n}(F)$的主块$K_{2n}(F)_l$，并分析了$K_{2n}(F)_l$中可分元素的群$D(n)_l$的伽罗瓦模结构，对于所有的$n>0$。如果$n$是奇数且与$l$互质，$F = K$是一个相当一般的全实数域，我们研究了$D(n)_l$的循环性与岩泽关于分圆域类群的经典猜想的关系，并将其推广到更广泛的数域类.最后，如果$F$是CM，我们的$l$ -adic Hecke特征标的特殊值被用来构造欧拉系统在奇数$K$ -群与系数$K_{2n+1}(F, \Bbb Z/l^k)$，对所有的$n>0$。这些都是Kolyvagin的高斯和欧拉系统的$K$理论欧拉系统在Banaszak-Gajda构造时，$K = \Bbb Q$。

英文摘要：

Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for $F$ with infinite type equal to a special value of certain $G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these characters, for primes $l> 2$. Further, the special values of these $l$-adic Hecke characters are used to construct $G(F/K)$-equivariant Stickelberger-splitting maps in the $l$-primary Quillen localization sequence for $F$, extending the results obtained in 1990 by Banaszak for $K = \Bbb Q$. We also apply the Stickelberger-splitting maps to construct special elements in the $l$-primary piece $K_{2n}(F)_l$ of $K_{2n}(F)$ and analyze the Galois module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$, for all $n>0$. If $n$ is odd and coprime to $l$ and $F = K$ is a fairly general totally real number field, we study the cyclicity of $D(n)_l$ in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if $F$ is CM, special values of our $l$-adic Hecke characters are used to construct Euler systems in the odd $K$-groups with coefficients $K_{2n+1}(F, \Bbb Z/l^k)$, for all $n>0$. These are vast generalizations of Kolyvagin's Euler system of Gauss sums and of the $K$-theoretic Euler systems constructed in Banaszak-Gajda when $K = \Bbb Q$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文