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Zbl 0681.12005
Gross, Benedict H.
On the values of abelian L-functions at $s=0$. (English)
[J] J. Fac. Sci., Univ. Tokyo, Sect. I A 35, No.1, 177-197 (1988).

Let $\zeta(s)$ be the zeta function of a global field k, then classically $\zeta(s)\equiv -hR\omega\sp{-1}s\sp{r\sb 1+r\sb 2-1}$ modulo $s\sp{r\sb 1+r\sb 2}$ near $s=0$. The author presents a conjecture similar to that. \par Let S be a finite set of places of k containing archimedean places, and let $A$ denote the $S$-integers, $U=A\sp*$, $h=\#Pic(A)$, $n=\#S-1$, $\omega =\#tors(U)$.The $S$-regulator $R$ is the absolute value of the determinant of $\lambda\sb{{\bbfR}}: U\to {\bbfR}\otimes X,\quad \epsilon \mapsto \sum\sb{s}\log \Vert \epsilon \Vert\sb v\cdot v,$ where $X=\{\sum a\sb vv:\sum a\sb v=0\}.$ Then one has $\zeta (s)\equiv -hR\omega\sp{- 1}s\sp n \pmod {s\sp{n+1}}.$ Let T be a finite set of places disjoint from S, and let $\zeta\sb T(s)$ be $\prod (1-N{\frak p}\sp{1-s})\zeta (s) $ for ${\frak p}\in T$, $U\sb T$ the units $\equiv 1 \pmod T,$ $Pic(A)\sb T$ the group of invertible A-modules with a trivialization at T. Making $\omega\sb T=\omega \cap U\sb T=1$ by restricting T, one thus obtains an integral formula $\zeta\sb T(s)\equiv m\cdot \det\sb{{\bbfR}}(\lambda)s\sp n \pmod{s\sp{n+1}},$ where $m=\pm h\sb T=\pm \#Pic(A)\sb T.$ \par To generalize the regulator, the author denotes by A the adeles of k, G a finite group, $f: A\sp*\to G$ a homomorphism, and defines $\lambda\sb G: U\to G\otimes X,\quad \epsilon \mapsto \sum\sb{s}f(1,...,\epsilon\sb v,...,1)v.$ Let $I=$ $\{\sum\sb{G}m(g)g :\sum m(g) =0\}\subset {\bbfZ}[G],$ then $G\cong I/I\sp 2$ (g$\mapsto g-1)$. So he defines $\det\sb G\lambda =\det ((g\sb{ij}-1))\in I\sp n/I\sp{n+1}$. Now let $\theta\sb G$ be the unique element of ${\bbfC}[G]$ such that $\theta\sb G(\chi)=L\sb T(\chi,0)$ for all characters $\chi\in \hat G$, where $L\sb T(\chi,s)$ is the abelian L-function; the author proves that $\theta\sb G$ is in ${\bbfZ}[G]$. He then states his conjecture: $\theta\sb G \equiv m\cdot \det\sb G \lambda \pmod{I\sp{n+1}}$. If the conjecture holds for $S,T,$ then it holds for $S'\supset S$, $T'\supset T$. Taking $G$ to be the Galois group of the maximal abelian extension of k unramified outside $S$ and $f$ the reciprocity map of global class field theory, one obtains a conjecture which implies all others. Then in sectionsd 5--6, the author proves the conjecture respectively for the number field $k$ and for $G\cong\bbfZ/\ell\bbfZ$ with prime $\ell$ up to a unit $u$ in $(\bbfZ/\ell\bbfZ)\sp*$. He finally discusses a refinement of Stark's conjecture for the first derivative [cf. {\it H.M. Stark}, Adv. Math. 35, 197--235 (1980; Zbl 0475.12018)]\ and the $L$-functions of quadratic characters.
[Zhang Xianke]

MSC 2000:: *11R42 Zeta functions and L-functions of global number fields
11R56 Adele rings and groups
11R58 Arithmetic theory of algebraic function fields

Keywords: values of abelian L-functions at $s=0$; zeta function; regulator; adeles

Citations: Zbl 0475.12018

Cited in: Zbl pre05302415 Zbl 1126.11066 Zbl 1149.11320 Zbl 1071.11064 Zbl 1067.11072 Zbl 1149.11319 Zbl 1057.11053 Zbl 1018.11059 Zbl 0903.11027 Zbl 0844.11071 Zbl 0820.11069 Zbl 0689.12002

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