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.
附加到数域的阿贝尔扩张上的Stickelberger元在一定条件下，在涉及高阶代数K-群的零化子关系中，是绝对参与的。在[Victor P. Snaith，Stark's conceptuation and new Stickelberger phenomenon，Canad.J.Math.58（2）（2006）419- 448]中，Snaith介绍了希望出现在零化子关系中的规范伽罗瓦模，推广和改进了涉及Stickelberger元素的那些规范伽罗瓦模。在本文中，我们研究了这些模块，对应于经典的Stickelberger元素，并证明了在特殊情况下与斯塔克单位的连接。
We exhibit an explicit short basis of the Stickelberger ideal of cyclotomic fields of any conductor $m$, i.e., a basis containing only short elements. By definition, an element of $\mathbb{Z}[G_m]$, where $G_m$ denotes the Galois group of the field, is called short whenever it writes as $\sum_{σ\in G_m} \varepsilon_σσ$ with all $\varepsilon_σ\in\{0,1\}$. One ingredient for building such a basis consists in picking wisely generators $α_m(b)$ in a large family of short elements. As a direct practical consequence, we deduce from this short basis an explicit upper bound on the relative class numbe…
Let $k$ be a real abelian number field and $p$ an odd prime not dividing $[k:\mathbb{Q}]$. For a natural number $d$, let $E_d$ denote the group of units of $k$ congruent to $1$ modulo $d$, $C_d$ the subgroup of $d$-circular units of $E_d$, and $\mathfrak{C}(d)$ the ray class group of modulus $d$. Let $ρ$ be an irreducible character of $G=\mathrm{Gal}(k/\mathbb{Q})$ over $\mathbb{Q}_p$ and $e_ρ \in \mathbb{Z}_p[G]$ the corresponding idempotent. We show that if the ramification index of $p$ in $k$ is less than $p-1$, then $|e_ρ \mathrm{Syl}_p(E_d/C_d) | = |e_ρ \mathrm{Syl}_p(\mathfrak{C}_d)|$ wh…
We generalize some results of Greither and Popescu to a geometric Galois cover $X\rightarrow Y$ which appears naturally for example in extensions generated by $\mathfrak{p}^n$-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz-Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in \cite{ABBL}) with Goss $ζ$-function.
In this paper we introduce confluence relations for motivic Euler sums (also called alternating multiple zeta values) and show that all linear relations among motivic Euler sums are exhausted by the confluence relations. This determines all automorphisms of the de Rham fundamental groupoid of $\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$ coming from the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$. Moreover, we also discuss other applications of the confluence relations such as an explicit $\mathbb{Q}$-linear expansion of a given motivic Euler sum by their basis…
Given a Galois cover of curves over $\mathbb{F}_p$, we relate the $p$-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a `weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of $p$-adic valuations of epsilon constants.
Class Number Relations in Abelian Extensions of Global Fields
University Marie et Louis Pasteur（玛丽和路易·巴斯德大学）；Lab. of Math. of Besançon（贝桑松数学实验室）；University of Ottawa（渥太华大学）；Department of Mathematics and Statistics（数学与统计系）
Let $Λ$ (isomorphic to $\mathbb{Z}_p[[T]]$) denote the usual Iwasawa algebra and $G$ denote the Galois group of a finite Galois extension $L/K$ of totally real fields. When the non-primitive Iwasawa module over the cyclotomic $\mathbb{Z}_p$-extension has a free resolution of length one over the group ring $Λ[G]$, we prove that the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive $p$-adic $L$-function (which is an element of a $K_1$-group) in a maximal $Λ$-order. This integrality result involves a careful study of the Dieudonné det…
